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Mccth iff'^y"^
i
HARVARD COLLEGE
SCIENCE CENTER
LIBRARY
SlOBrtiiulwiiy
laHmammmm
FACTOR TABLE
FOB THE
SIXTH MILLION,
CONTAINING THB
LEAST FACTOR OF EVERY NUMBER NOT DIVISIBLE BY 2, 3, OR 5
BETWEEN
5,000,000 AND 6,000,000.
BY
JAMES GLAISHER, F.R.S.
LONDON:
TAYLOR AND FRANCIS, RED LION CXDURT, FLEET STREET.
1883
:l ^
I
Math l\tn-4
HARVARD ,
UNIVERSITY
LIBRARY
DEC 9 1963
ALB1B
FLAMMAM.
PRINTED BY TAYLOR AND FRANCIS^
BED UON COURT, FLEET STREET.
PREFACE.
The table contained in the present volume is similar in every respect to those
contained in the two preceding volumes. It gives the least factor of every number
between 5,000,000 and 6,000,000, and completes the series of factor tables begun by
Burckhardt and continued by Dase.
The gap of three millions between the third million and the seventh million
being now filled in, the tables extend from unity to 9,000,000. The dates of
publication of the nine millions are: — Second, 1814; Third, 1816; First, 1817;
Seventh, 1862; Eighth, 1863; Ninth, 1865; Fourth, 1879; Fifth, 1880; Sixth, 1883.
A full description of the mode of construction of the table, and a historical
account relating to factor tables and to the distribution of primes, formed part of
the Introduction to the Fourth Million.
The results of the enumeration of the primes in the fourth and fifth millions
were given in the Introductions to those Millions. The present Introduction
contains the detailed results of the enumeration of the primes in the whole nine
millions^ with a comparison between the actual numbers and the values given by
the various formulae which have been proposed for the approximate representation
of the numbers of primes between given limits. The concluding portion relates to
the function li x and to the calculation of Riemann's formula.
JAMES GLAISHER.
Blaokheath, London,
May 16, 1883.
b2
i
CONTENTS OF THE INTRODUCTION.
Paok
Section I. Manneb of using the Table 7
Section II. Enumeration of the Prime Numbers in the First Nine Millions . . 7
Specimen of the forms used in enumerating the Primes ... 8
Table I. — Number of Primes in each successive group of 1000 numbers,
from to 9,000,000 10
Table II. — Number of Primes in each successive group of 10,000
numbers, from to 9,000,000 28
Table showing the number of Primes in each successive group of
100,000 numbers, from to 9,000,000 30
Table showing the number of Primes in each successive group of
250,000 numbers, from to 9,000,000 31
Table showing the number of Primes in each successive group of
500,000 numbers, from to 9,000,000 31
Table showing the number of Primes for the complete Millions . 31
Table showing the total number of Primes in the first Million, in the
first two Millions, in the first three Millions, &c. . . . 32
Enumeration according to Numbers of Primes in Centuries . 32
Tables showing the numbers of Centuries which contain no prime, one
prime, two primes, three primes, &c., in each group of 10,000
numbers from to 9,000,000 33
Similar tables for the complete Millions :
First Million 56
Second and Third Millions 57
Fourth and Fifth Millions 58
Sixth and Seventh Millions 59
Eighth and Ninth Millions 60
Final results for the nine Millions 61
List of the Centuries in the nine Millions which consist wholly of
Composite numbers 61
6
CONTENTS OF THE INTEODUCTION.
Section II.
(continued.)
Secttion III.
Sequences of Composite Numbers :
Sequences of 81 and upwards in the Sixth Million .
Sequences exceeding 100 in the nine Millions ....
Sequences exceeding 130 in the nine Millions ....
The two longest Sequences in each of the nine Millions
COMPABISON OF THE RESULTS OF THE ENUMERATION WITH LeGENDRE's, TcHEBY-
cheff's, and Riemann's Formula
Legendre's Formula .
TchebycheflF's Formula
Comparison of the Numbers of Primes with Legendre's and TchebycheflTs
Formulae ...........
Table I. — Comparison with Legendre's and TchebycheflTs Formulae at
intervals of 50,000 up to 9,000,000
Table II. — Comparison with Legendre's and TchebychefF's Formulae for
groups of 50,000 up to 9,000,000
Table III. — Value of the constant A in Legendre's Formula determined
at intervals of 50,000 up to 9,000,000
Approxiniations to Tchebycheffs Formula
Table IV.— Comparison with i^g^Jl.os3QQ '
■i and
logo:— 1 —
log 57—1
log^
at intervals of 50,000 up to 9,000,000
Riemann's Formula
Table V. — Comparison with Riemann's, Tchebycheff's, and
Formulae at intervals of 50,000 up to 9,000,000 .
Table VI. — ^Deviations of the Five Formulae
Number of Primes inferior to 10,000,000 and to 100,000,000
On the Calculation of the Function \ix
On the Calcidation of Riemann's Formula .
Legendre's
Paob
62
64
65
65
66
66
69
70
71
74
77
79
80
83
84
87
89
89
99
Diagram showing the deviations of the five Formulae
facing page 88
{
I
INTRODUCTION.
SECTION I.
MANNER OF USING THE TABLE.
The object of the Table is to resolve into factors any one of the million numbers lying between
5,000,000 and 6,000,000; and the Table gives directly the least factor of every number not divisible
by 2, 3, or 5 between these limits.
In entering the Table, the first three figures are to be found in the spaces above the headings
of the columns, the next two among the headings of the columns, and the last two in the column
at the left-hand side of the page. Thus to find the least factor of 5,004,763, enter the Table with
500 47 63 (on the first page of the Table) and the tabular result is 491, showing that 491 is the
least &ctor of the number.
In the case of prime numbers, a bar is printed as tabular result : thus, for example, 5,004,767
is a prime.
If the given number be divisible by 2, 3, or 5, it will not be found in the Table. The least
factors of such numbers may be determined by inspection if they are divisible by 2 or 5, and
almost by inspection if 3 is the least factor.
The first three figures of the arguments are printed in the first column of tabular results, on
each page, and also whenever on the page a change occurs in these figures.
To facilitate the entry of the Table, these leading figures are also printed at the outside
corner of every page, in square brackets. The number printed on each page as a heading is the
first of the 9000 numbers to which the page relates ; these headings are not really required, as
the Table is most conveniently entered by means of the leading figures in the comer ; but they are
retained for the sake of uniformity with the six millions of Burckhardt and Dase.
The tables of Burckhardt, Dase, and Chemac were described in Section II. of the Intro-
duction to the Fourth Million.
SECTION II.
ENUMERATION OF THE PRIME NUMBERS IN THE FIRST NINE MILLIONS.
Thb enumeration was made in the following manner : — ^The number of primes in each group of
100 numbers, or century, was found by counting from the tables and entered in its place
upon a printed form. This form contained ten lines and ten columns, so that the numbers
8
INTEODUCTION.
[§II.
found by adding up the results in each column were the numbers of primes in the successive
groups of 1000 numbers, and the sum of the ten numbers at the foot of the columns was the
number of primes in the group of 10,000 to which the form related. I reproduce as specimens
the blank form and the fiUed-in forms corresponding to the first two groups of 10,000.
No, Number op Pbijces =»
00
10
20
30
40
50
60
70
80
90
01
11
21
31
41
51
61
71
81
91
02
12
22
32
42
52
62
72
82
92
03
13
23
33
43
53
63
73
83
93
04
14
24
34
44
54
64
: 74
84
94
05
15
2b
35
45
55
65
*
i 75
i
85
95
06
16
26
36
46
56
66
76
86
96
07
17
27
37
47
57
67
77
87
97
08
18
28
38
48
58
68
78
88
98
09
19
29
39
49
59
69
79
89
99
m
No.O
Number of Frtmbs
= 1230.
•
00
26
10
16
20
14
30
12
40
IS
50
12
60
12
70
9
80
II
90
II
01
21
11
12
21
10
31
10
41
9
61
II
61
II
71
10
81
10
91
12
02
16
12
15
22
IS
32
II
42
16
52
10
62
13
72
II
82
14
92
11
03
16
13
II
23
IS
33
IS
43
9
53
10
63
IS
73
9
83
9
93
II
04
17
14
17
24
10
34
II
44
II
54
13
64
8
74
II
84
8
94
IS
05
14
15
12
25
II
35
14
45
12
55
13
65
II
75
IS
85
12
95
7
06
16
16
15
26
IS
36
13
46
12
56
12
66
10
76
12
86
13
96
13
07
14
17
12
27
14
37
12
47
12
57
10
67
12
77
10
87
II
97
II
08
IS
18
12
28
12
38
II
48
8
58
16
68
12
78
10
88
13
98
12
09
14
19
13
135
29
II
39
II
120
49
IS
59
7
114
69
13
79
10
89
*
9
no
99
9
169
127
119
117
107
112
No.l
Number of Primes
= 1033.
00
II
10
10
20
9
30
II
40
10
50
9
60
12
70'
II
80
10
90
II
01
12
11
II
21
II
31
12
41
8
51
12
61
9
71
9
81
II
91
7
02
10
12
10
22
12
32
9
42
7
52
12
62
8
72
9
82
II
92
II
03
12
13
10
23
9
33
10
43
9
53
12
63
9
73
12
83
II
93
9
04
10
14
II
24
13
34
II
44
12
54
II
64
12
74
13
84
II
94
16
05
8
15
9
25
12
35
8
45
12
55
8
65
7
75
10
85
10
95
II
06
12
16
8
26
12
36
12
46
10
56
13
66
13
76
8
86
6
96
7
07
II
17
9
27
10
37
12
47
14 1
1
57
12
67
7
77
10
87
II
97
10
08
10
18
12
28
9
38
9
48
12
58
9
68
9
78
8
88
S
98
10
09
10
19
13
29
12
39
II
loS
49
8
t
59
10
69
12
98
79
14
104
89
8
94
99
12
106
103
109
1
102
108
104
§ II.] ENTTMEBATIOy OP PBIMES IN THE FIEST NINE MILLIONS. 9
The first of these two tables shows that the number of primes between and 100 is 26,
between 100 and 200 is 21, and so on, and that the numbers of primes in the successive groups
of 1000 numbers, commencing with 0, are 169, 135, &c., and that the total number of primes in
the 10,000 numbers or myriad is 1230, this being the number entered at the top of the form.
The second table shows that the number of primes between 10,000 and 10,100 is 11,
between 10,100 and 10,200 is 12, and so on, and that the numbers of primes in the successive
groups of 1000 numbers, starting from 10,000, are 106, 103, &c., and that the total number of
primes in the myriad 10,000 to 20,000 is 1033*.
The enumeration for the whole nine millions has been performed in this manner ; and from
the 900 tables, similar to the specimens, all the results contained in this Section, excepting only
the sequences, have been derived.
The enumerations were made, and the results entered upon the forms, at least twice, and
for most of the millions three times, independently. They were not only made at different
times (between 1871 and 1881) and by different computers, but they have all been carefully
examined since, and I feel no doubt of their accuracy. In the case of the fourth, fifth, and sixth
millions the enumerations were made both from the manuscript and from the stereotype pages.
Table I. (pp. 10-27) gives the number of primes in each successive group of a thousand
numbers from to 9,000,000. Thus, for example, it shows that the number of primes between
and 1000 is 169, between 2000 and 3000 is 135, . . . between 50,000 and 51,000 is 89, and so
on. Each pair of columns contains a hundred arguments corresponding to the hundred groups
of 1000, which form the group of 100,000 which appears as a heading ; so that, for example,
between the numbers 3,364,000 and 3,365,000 there are 74 prinaes (p. 16). Each million
occupies two pages. The results in this table are the numbers which appear at the foot of
the columns in the 900 tables entered upon the printed forms ; thus the first twenty results in
the first million (p. 10) are the twenty numbers given at the foot of the columns in the two
specimen-tables on the opposite page. The sums of the numbers forming each column of
results are given at the foot of the column ; these sums are therefore the numbers of primes
in the groups of 50,000 to which the columns relate.
The only table similar to Table I. which has been published, occurs in the second volume
of Gauss's Werkey and relates to the first million only. An account of this table and of other
tables relating to an enumeration of the primes in the second and third millions made by
Goldschmidt at Gauss's suggestion, which appear in the same volume, was given on pp. 31, 32 of
the Introduction to the Fourth Million. When the nineteen errors found by Meissel and the
error in the 354th chiliad, which are there referred to, are corrected, the results agree with those
given on pp. 10 and 11.
Table II. (pp. 28 and 29) gives the numbers of primes in each group of 10,000 from
to 9,000,000, and is similar in arrangement to Table I. Thus, for example, the number of
primes between and 10,000 is 1230, between 10,000 and 20,000 is 1033, and so on. Each pair
of columns therefore corresponds to a million numbers. The results in this table are the numbers
entered at the top of each of the printed forms ; thus, for example, the first result in the first
million is 1230 and the second is 1033, the numbers which appear at the top of the two
specimen-tables.
The results in this Table which relate to the first million and the last four millions are
new. Those relating to the fourth and fifth millions have been published in the Introductions
to these Millions. Similar results for the second and third millions occur in Gauss, but 45 out
of the 200 numbers there given are incorrect. The Tables on pp. 30 and 31 give the numbers
of primes in each group of 100,000, of 250,000, of 500,000, and of 1,000,000 respectively.
Differences are added, as it is interesting to observe the fluctuations in the numbers of primes
in these more extended groups. It will be noticed that not even in the table where the groups
contain 500,000 numbers do the numbers of primes in the successive groups always diminish,
there being an increase in one instance. The results in these tables were obtained by addition
from the numbers at the foot of the columns in Table 1.
* In geotion YII. of the Introduction of the Fourth Million (p. 30) a similar specimen-table rekting to thp nun^l^rs
between 3,000,000 and 3,010,000 was given.
10
INTEODUCTION.
Wn.
TABLE I.
NUMBBB OF FbIMES IN EACH SUCCESSITE OBOUP OF 1000 NuMBEBS, FBOM TO 9,000,000.
FiBST Million.
100,000
200,000
300,000
400,000
to
to
to
to
to
100,000
200,000
300,000
•
400,000
600,000
o
169
SO
89
81
50
85
77
50
71
85
50
z*
70
50
9;
I
"35
5»
97
r
93
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?°
I
87
5«
88
I
83
51
81
I
7"
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76
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127
5»
89
, 2
87
5*
88
2
78
5»
78
2
7»
5a
82
2
76
5a
«3
3
120
53
9»
3
80
53
77
3
78
53
81
3
84
53
76
3
75
53
7a
4
119
54
90
4
9«
54
84
4
77
54
76
4
88
54
87
4
70
54
74
5
114
55
93
5
82
55
85
5
83
55
87
5
80
55
79
5
83
55
82
6
"7
56
99
6
9»
56
76
6
83
56
7a
6
82
56
67
6
67
56
73
7
107
57
91
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57
88
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57
78
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57
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57
77
8
no
58
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58
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9
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59
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9
88
59
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9
88
59
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9
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59
68
lO
106
60
88
10
83
60
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60
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10
73
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103
61
87
II
84
61
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86
61
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69
61
79
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81
61
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109
61
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63
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102
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64
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108
65
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88
80
38
87
88
77
38
77
88
75
38
75
88
83
39
96
89
94
39
84
89
88
39
79
89
85
39
68
89
84
39
75
89
78
40
88
90
89
40
87
90
75
40
80
90
84
40
84
90
81
40
79
90
78
4«
lOI
9*
85
41
87
9»
81
4"
9«
9»
77
4«
77
9"
79
4»
Z*
9>
76
4*
102
92
^l
42
82
92
89
4»
76
92
78
42
77
92
86
4a
85
9a
67
43
85
93
86
43
77
93
84
43
77
93
68
43
80
93
87
43
88
93
82
44
96
94
87
44
79
94
74
44
88
94
85
44
80
94
75
44
82
94
80
45
86
95
95
45
85
95
85
45
80
95
75
45
76
95
7*
45
68
95
87
46
90
96
84
46
84
96
76
46
84
96
82
46
80
96
75
46
68
96
68
47
95
97
82
47
83
97
87
47
79
97
73
47
82
97
75
47
73
97
81
48
89
98
87
48
83
98
86
48
88
98
73
48
77
98
82
48
70
98
72
49
98
99
87
49
9»
99
77
49
80
99
78
49
82
99
81
49
80
99
81
Total
5*34
4459
4256
4136
4060
3953
3980
3883
3846
3832
\
$n.]
ENUMEBATION OP PRIMES IN THE FIEST NINE MILLIONS.
11
TABLE I. (continued).
NxnaiBR OF Fbimss in each successive gboup of 1000 Nttubebs, from to 9,000,000.
FiBST Million.
600,000
600,000
700,000
800,000
900,000
to
to
to
to
to
600,000
700,000
800,000
900,000
1,000,000
o
79
50
79
76
50
61
75
50
68
85
50
70
74
50
76
z
v^
5«
75
I
73
51
74
I
71
5'
85
I
66
51
77
I
73
5»
70
2
67
5*
71
2
83
5a
85
2
81
5a
73
2
70
5a
7A'
2
70
5a
78
3
76
53
80
3
76
53
69
3
71
53
71
3
69
53
66
3
63
53
«5
4
7«
54
77
4
73
54
78
4
87
54
83
4
78
54
71
4
81
54
73
5
^1
55
61
5
74
55
73
5
68
55
70
5
79
55
73
5
70
55
76
6
76
56
88
6
7»
56
71
6
82
56
66
6
68
56
7!
6
80
56
58
7
71
57
68
7
78
57
70
7
74
57
68
7
70
57
76
7
80
57
69
8
76
58
74
8
78
58
79
8
77
58
79
8
69
58
69
8
Z9
S8
77
9
75
59
77
9
80
59
73
9
77
59
77
9
78
59
71
9
82
59
69
lO
72
60
86
10
73
60
83
10
78
60
77
10
78
60
77
10
62
60
68
II
82
61
61
II
7»
61
70
II
76
61
80
II
7a
61
74
II
81
61
88
12
70
62
83
12
76
62
74
12
7a
62
68
12
69
62
83
12
71
62
71
13
77
63
67
13
79
63
77
»3
73
63
79
13
7a
63
60
13
54
63
74
»4
81
64
77
»4
71
64
77
H
66
64
7a
H
78
64
80
14
73
64
74
15
66
65
78
"5
75
65
77
15
83
65
82
15
69
65
80
"5
70
65
70
16
85
66
72
16
85
66
73
16
69
66
78
16
75
66
68
16
72
66
73
17
83
67
7*
17
81
67
74
«7
65
67
68
17
75
67
78
17
79
67
66
18
76
68
7«
18
67
68
66
18
67
68
77
18
62
68
80
18
75
68
75
19
78
69
80
^9
73
69
74
19
74
69
74
19
83
69
73
19
71
69
73
ao
73
70
85
20
77
70
75
20
78
70
77
20
75
70
79
20
72
70
76
21
83
71
72
21
70
7>
76
21
77
71
75
21
7»
71
5?
21
7a
71
78
22
79
7»
85
22
74
7»
76
22
73
72
70
22
84
7a
76
22
7a
72
74
a3
69
73
7»
»3
75
73
77
^3
86
73
76
23
78
73
65
a3
81
73
P
«4
77
74
70
44
4
78
74
73
24
75
74
7a
a4
71
74
75
H
76
74
85
as
79
75
77
*5
69
75
75
as
69
75
67
a5
81
75
80
a5
80
75
70
20
84
76
78
26
70
76
74
26
76
76
70
26
78
76
75
26
74
76
66
27
72
77
77
»7
70
77
63
27
75
77
76
27
69
77
67
27
63
77
60
28
70
78
76
28
71
78
82
28
76
78
81
28
69
78
78
28
70
78
80
29
78
79
77
*9
75
79
83
29
75
79
71
29
68
79
75
29
80
79
65
30
80
80
73
30
67
80
75
30
69
80
70
30
76
80
80
30
69
80
67
31
. 68
81
79
31
81
81
78
31
76
81
82
3«
79
81
69
31
69
81
75
3a
79
82
73
32
77
82
66
3*
71
82
68
3a
82
82
7a
3a
76
82
70
33
74
83
78
33
70
83
78
33
75
83
75
33
68
83
l^
33
68
83
70
34
7*
84
7a
34
82
84
72
34
74
84
75
34
67
84
69
34
81
84
70
35
71
85
81
35
78
85
74
35
79
85
77
35
73
85
77
35
68
85
76
36
87
86
79
36
73
86
74
36
69
86
70
36
71
86
76
36
71
86
76
37
67
87
87
37
74
87
82
37
78
87
73
37
64
87
71
37
71
87
63
38
78
88
73
38
59
88
74
38
70
88
80
38
80
88
77
38
7a
88
71
39
71
89
68
39
7*
89
79
39
81
89
68
39
69
89
68
39
68
89
7a
40
73
90
71
40
77
90
60
40
67
90
78
40
70
90
68
40
74
90
71
41
77
9»
67
4«
71
91
79
4»
74
91
71
4*
69
91
80
41
79
91
79
4a
78
9*
80
42
68
92
77
4a
73
92
82
4a
83
9a
69
4a
7a
9a
!2
43
81
93
77
43
70
93
73
43
67
93
71
43
68
^3
72
43
76
93
68
4f
68
94
78
44
86
94
76
44
64
94
73
44
78
94
74
44
73
94
7i
45
69
95
77
45
75
95
77
45
67
95
79
45
70
95
80
45
66
95
69
46
73
96
79
46
74
96
73
46
76
96
77
46
69
96
64
46
72
96
69
47
76
97
73
47
79
97
79
47
71
97
72
47
77
97
75
47
66
97
83
48
77
98
7a
48
73
98
62
48
76
98
71
48
75
98
76
48
67
98
74
49
78
99
73
49
84
99
7*
49
7a
99
71
49
68
99
61
49
75
99
65
Total
3784
3776
3733
3712
3695
3713
3666
3657
3633
3591
c2
1%
INTEODTJCTION.
K 11:
TABLE 1. (contirmed).
Number of Fbimes is each successive group of 1000 Numbers, from to 9,000,000.
Second Million.
1,000,000
1,100,000
1,200,000
1,300,000
1,400,00C
1
to
to
to
to
to
1,100,000
1,200,000
1,300,000
1,400,000
1,600,00€
\
o
75
50
62
73
50
77
64
50
70
71
50
75
!l
50
75
I
77
51
73
I
71
51
75
I
68
51
64
I
74
5«
77
I
66
51
64
2
81
5»
67
2
78
5»
65
2
74
5a
69
2
66
5a
s?
2
64
5a
70
3
72
53
67
3
68
53
7a
3
7*
53
62
3
82
53
66
3
76
53
72
4
63
54
73
4
77
54
70
4
61
54
79
4
66
54
71
4
63
54
77
5
73
55
70
5
76
55
75
5
66
55
78
5
64
55
79
5
66
55
82
6
82
56
73
6
75
56
74
6
70
56
67
6
77
56
75
6
65
56
61
7
76
57
66
7
67
57
77
7
65
57
72
7
65
57
56
7
76
57
69
8
79
58
76
8
83
58
75
8
64
58
82
8
70
58
81
8
70
58
68
9
75
59
71
9
68
59
68
9
7a
59
70
9
74
59
64
9
68
59
63
lO
66
60
70
10
76
60
70
10
76
60
74
10
74
60
74
10
75
60
74
II
70
61
7*
II
74
61
80
II
75
61
65
II
74
61
70
II
67
61
78
la
83
62
59
12
81
62
81
12
75
62
72
12
71
62
69
12
81
62
76
13
67
63
84
13
68
63
82
13
76
63
75
"3
64
63
74
"3
67
63
68
»4
71
64
69
14
7»
64
68
14
76
64
66
14
64
64
68
H
57
64
68
15
74
65
71
15
74
65
73
15
67
65
80
15
75
65
68
15
62
^J
79
i6
7*
66
69
16
62
66
70
16
7a
66
79
16
69
66
75
16
68
66
70
17
7»
67
73
17
79
67
73
17
80
67
68
17
74
67
68
17
70
67
69
i8
70
68
73
18
72
68
69
18
66
68
69
18
76
68
74
18
62
68
61
'?
74
69
76
19
58
69
68
"9
81
69
70
19
62
69
69
»9
72
69
73
20
79
70
74
20
82
70
76
20
60
70
74
20
65
70
7|
20
71
70
58
21
74
7"
75
"21
65
71
70
21
67
71
77
21
80
71
68
21
68
7»
74
22
7»
7»
64
22
64
72
71
22
70
72
72
22
84
72
68
22
65
7a
75
13
70
73
69
«3
75
73
63
H
69
73
73
23
71
73
66
a3
80
73
68
»4
70
74
75
»4
67
74
76
H
69
74
60
a4
69
74
71
a4
71
74
67
*5
78
75
78
»5
73
75
73
*5
77
75
67
a5
65
75
75
a5
69
75
5?
26
73
76
75
26
67
76
69
26
75
76
73
26
79
76
71
26
11
76
76
27
67
77
65
»7
76
77
73
27
61
77
66
a7
59
77
76
a7
77
77
28
74
78
66
28
79
78
82
28
75
78
68
28
75
78
81
28
65
78
73
29
75
79
81
29
70
79
73
29
70
79
79
29
66
79
74
29
75
79
78
30
71
80
82
30
70
80
83
30
65
80
64
30
64
80
70
30
66
80
70
31
62
81
71
31
79
81
69
31
85
81
73
31
76
81
77
31
78
81
69
3*
72
82
68
3a
71
82
70
3a
61
82
79
3a
73
82
60
3a
7a
82
70
33
75
83
73
33
74
83
76
33
62
83
66
33
71
83
63
33
7a
53
75
34
78
84
71
34
71
84
75
34
64
84
67
34
62
84
68
34
76
84
68
35
66
85
66
35
75
85
74
35
72
85
77
35
66
^1
81
35
73
l^
62
36
78
86
75
36
73
86
62
36
60
86
73
36
80
86
71
36
69
86
75
37
65
87
70
37
69
87
80
37
77
87
77
37
70
87
65
37
75
87
69
38
70
88
74
38
67
88
7»
38
73
88
78
38
69
88
68
38
74
88
68
39
64
89
56
39
70
89
74
39
74
89
68
39
- 74
89
7a
39
68
89
80
40
77
90
74
40
67
90
68
40
68
90
64
40
75
90
75
40
67
90
75
41
7A-
91
79
41
73
91
74
41
73
91
66
41
69
91
67
41
73
91
7«
4»
70
9»
76
4*
70
92
73
42
67
92
73
42
78
92
62
4a
68
92
71
43
73
93
81
43
65
93
76
43
71
93
69
43
73
93
80
43
68
93
75
44
77
94
79
4*
76
94
71
44
77
94
63
44
66
94
68
44
69
94
74
45
72
95
7»
45
67
95
65
45
63
95
73
45
69
95
7a
45
7a
95
68
46
65
96
68
46
70
96
68
46
80
96
70
46
67
96
81
46
7a
96
70
47
78
97
72
47
75
97
7*
47
72
97
78
47
67
97
74
47
79
97
69
48
66
98
60
48
65
98
72
48
76
98
60
48
71
98
79
48
60
98
$5
49
79
99
77
49
7»
99
74
49
77
99
73
49
57
99
83
49
74
99
60
Total
3636
3580
3589
3636
3530
3551
3522
3581
m
3502
3526
§n.]
ENUMBEATION OF PEIMES IN THE FIBST NINE MILLIONS.
la
TABLE I. (continued).
NCMBBS OF FbIMES IK EACH SUCCESSIVE GROUP OF 1000 NUMBEBS, FROM TO 9,000,000.
Second Million.
1,500,000
1,600,000
1,700,000
1,800,000
1,900,000
to
to
to
to
to
1,600,000
•
1,700,000
1,800,000
1,900,000
2,000,000
►
o
83
50
72
72
50
75
68
50
72
75
50
57
69
50
67
I
69
5*
67
I
77
51
63
I
81
51
69
I
64
51
82
I
63
5»
68
£
68
5»
68
2
79
5a
74
2
72
5*
81
2
60
5*
68
2
!2
5*
68
3
75
53
73
3
77
53
71
3
56
53
71
3
70
53
69
3
68
53
7*
4
78
54
61
4
7*
54
73
4
70
54
64
4
71
54
78
4
68
54
78
5
76
55
84
5
70
55
75
5
67
55
67
5
70
55
78
5
64
55
67
6
76
56
69
6
72
56
76
6
71
56
69
6
77
56
66
6
69
56
!5
7
61
57
71
7
69
57
64
7
70
57
65
7
73
57
70
7
74
57
67
8
80
58
78
8
64
58
65
8
70
58
67
8
66
58
69
8
7*
58
61
9
65
59
71
9
67
59
65
9
71
59
59
9
78
59
70
9
73
59
71
lo
79
60
66
10
77
60
67
10
71
60
64
10
70
60
73
10
69
60
70
IZ
69
61
63
II
68
61
67
II
69
61
62
II
66
61
67
II
71
61
'6l
1%
63
62
58
12
65
62
78
12
66
62
64
12
68
62
64
12
74
62
»3
81
63
78
»3
59
63
67
13
62
63
70
13
65
63
77
13
70
63
65
14
68
64
70
14
74
64
56
«4
73
64
73
14
7*
64
66
»4
69
64
69
15
62
65
78
15
70
65
74
15
68
65
66
15
75
65
80
15
77
«5
66
i6
71
66
68
16
71
66
77
16
75
66
66
16
62
66
73
16
67
66
68
17
74
67
68
17
64
67
82
17
58
67
77
17
61
67
73
17
76
67
6z
i8
68
68
62
18
76
68
69
18
79
68
7"
18
68
68
70
18
65
68
68
19
67
69
70
»9
70
69
79
19
64
69
66
»9
65
69
60
«9
59
69
63
20
68
70
78
20
69
70
68
20
71
70
71
20
78
70
71
20
73
70
66
21
66
71
68
21
7a
71
60
21
70
71
62
21
62
71
71
21
71
71
58
22
61
72
7»
22
69
72
66
22
66
72
69
22
66
7*
65
22
58
72
73
23
77
73
72
23
69
73
75
»3
58
73
79
»3
P
73
65
23
75
73
71
*4
75
74
67
^4
67
74
72
24
75
74
7*
24
80
74
67
H
63
74
55
»5
73
75
69
25
74
75
76
*5
64
75
67
»5
68
75
74
as
73
75
66
26
58
76
72
26
75
76
66
26
79
76
70
26
77
76
67
26
78
76
78
»7
74
77
70
27
78
77
66
27
74
77
64
27
66
77
73
27
79
77
5l
28
65
78
60
28
69
78
83
28
63
78
80
28
73
78
75
28
68
78
66
29
74
79
73
29
69
79
70
29
71
79
66
29
69
79
61
*9
73
79
72
30
66
80
71
30
69
80
72
30
66
80
77
30
72
80
75
30
66
80
79
31
59
81
66
31
69
81
68
3»
64
81
69
31
77
81
7a
31
71
81
65
3»
67
82
65
3*
74
82
65
32
7»
82
71
32
66
82
60
3»
66
82
^9
33
69
83
7»
33
75
83
56
33
68
83
7*
33
69
83
72
33
64
!3
64
34
68
84
73
34
78
84
69
34
67
84
66
34
73
84
69
34
70
84
7«
35
76
85
66
35
68
85
79
35
68
85
65
35
61
85
73
35
66
85
61
36
70
86
66
36
79
86
69
36
70
86
7»
36
60
86
68
36
66
86
69
37
71
87
64
37
53
87
69
37
61
87
70
37
66
87
67
37
67
87
72
38
73
88
76
38
60
88
68
38
78
88
66
38
64
88
66
38
69
88
71
39
67
89
74
39
67
89
60
39
75
89
62
39
67
89
75
39
79
89
58
40
76
90
65
40
68
90
68
40
70
90
74
40
63
90
73
40
67
90
77
4»
67
91
61
4»
62
9"
74
4»
65
9"
7*
4*
69
91
72
41
68
91
71
4»
62
92
68
4a
73
92
71
42
67
92
71
42
78
92
74
4»
7a
92
69
43
64
93
64
43
75
93
67
43
68
93
70
43
72
93
69
43
73
93
7"
44
62
94
79
44
68
94
72
44
66
94
71
44
70
94
61
44
, 68
94
66
45
79
9|
' 73
45
73
95
68
45
71
95
77
45
66
95
68
45
76
9|
69
46
72
96
62
46
72
96
67
46
76
96
61
46
75
96
74
46
74
96
77
47
76
H
76
47
67
97
76
47
68
97
61
47
7*
97
67
47
67
97
75
48
73
98
68
48
5!
98
73
48
74
98
87
48
63
98
71
48
71
98
21
49
67
99
60
49
76
99
77
49
81
99
69
49
73
99
62
49
56
99
66
Total
3508
3465
3508
3507
3467
3465
3470
3487
3473
3430
14
INTEODTJCnON.
K "•
TABLE I. (continued).
NuMBEB OF Primes in each successive qboup op 1000 Numbebs, from to 9,000,000.
Third Million.
2.000,000
2,100,00c
f
2,200,000
2,300,000
2,400,000
to
to
to
to
to
2,100,000
•
2^00,000
2,300,00C
\
2,400,000
2,500,000
o
69
50
74
71
50
64
74
50
55
73
50
79
^J
50
60
I
66
51
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5«
69
I
64
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71
I
76
5'
|9
I
58
5»
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%
71
5»
65
2
7»
5*
76
2
70
5*
61
2
61
5a
69
2
69
5»
75
3
e?
53
68
3
66
53
67
3
70
53
78
3
68
53
69
3
75
53
70
4
70
54
7»
4
74
54
68
4
69
54
67
4
69
54
66
4
66
54
«7
5
73
55
60
5
70
55
69
5
65
55
7»
5
57
55
63
5
70
55
64
6
64
56
76
6
73
56
71
6
74
56
78
6
65 .
56
70
6
61
56
67
7
82
57
7«
7
63
57
80
7
75
57
71
7
75
57
Z^
7
70
57
63
8
74
58
73
8
74
58
80
8
64
58
74
8
76
58
67
8
61
58
75
9
69
59
7*
9
64
59
68
9
76
59
61
9
70
59
75
9
66
59
74
10
60
60
62
10
72
60
64
10
81
60
80
10
70
60
64
10
68
60
74
II
69
61
65
II
68
61
67
II
57
61
64
II
69
61
66
II
60
61
67
12
66
62
75
12
61
62
69
Z2
70
62
62
12
57
62
71
12
71
62
60
13
77
63
74
«3
68
63
74
13
59
63
7a
13
63
63
73
13
78
63
65
14
77
64
75
«4
68
64
58
14
58
64
76
H
69
64
75
H
70
64
82
«5
7*
65
63
15
74
65
69
15
74
65
60
«5
67
65
66
15
60
65
65
i6
69
66
74
16
77
66
59
16
67
66
76
16
63
66
69
16
73
66
62
«7
68
67
59
«7
65
67
66
17
64
67
70
17
63
67
69
17
77
67
66
i8
63
68
73
18
67
68
79
18
64
68
70
18
7'
68
69
18
66
68
65
»9
70
69
74
19
63
69
62
19
66
69
7«
»9
70
69
69
19
69
69
62
20
71
70
68
20
69
70
70
20
63
70
74
20
61
70
66
20
65
70
68
11
7*
71
69
21
66
71
65
21
67
71
76
21
62
71
63
21
73
71
7«
21
66
72
63
22
69
72
82
22
68
72
72
22
71
72
74
22
73
72
68
a3
73
73
66
a3
67
73
62
23
74
73
70
23
73
73
65
*3
62
73
66
14
59
74
66
14
63
74
6S
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65
74
74
H
59
74
56
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61
74
66
*5
69
75
70
45
71
75
67
a5
70
75
67
*5
60
75
68
*5
67
75
62
i6
67
76
59
26
76
76
77
26
72
76
61
26
76
76
73
26
73
76
69
27
73
77
68
17
69
77
71
27
75
77
68
27
81
77
64
*7
66
77
68
28
69
78
70
28
69
78
67
28
73
78
69
28
64
78
67
28
68
78
64
29
74
79
75
29
78
79
64
»9
68
79
63
29
65
79
66
^9
64
79
7<^
30
64
80
68
30
60
80
68
30
65
80
65
30
62
80
71
30
66
80
67
3»
62
81
63
3«
68
81
64
31
69
81
66
31
60
81
79
31
69
81
69
3»
86
82
67
32
73
82
65
3*
64
82
61
32
70
82
7»
3a
61
82
69
33
74
83
70
33
63
83
67
33
75
83
62
33
79
83
63
33
64
83
58
34
68
84
62
34
62
84
74
34
69
84
61
34
66
84
64
34
75
84
70
35
66
85
7*
35
69
85
66
35
75
85
75
35
69
85
64
35
71
8S
70
36
68
86
7*
36
79
86
68
36
68
86
64
36
70
86
73
36
64
86
77
37
65
87
73
37
7a
87
65
37
61
87
71
37
68
87
75
37
63
87
74
38
66
88
75
38
63
88
7»
38
70
88
69
38
63
88
68
38
60
88
68
39
71
89
64
39
64
89
70
39
64
89
68
39
64
89
76
39
64
89
65
40
68
90
65
40
73
90
71
40
70
90
7»
40
74
90
68
40
80
90
67
41
62
9>
72
4«
69
91
63
4"
68
91
67
4«
69
91
67
41
68
9"
63
4*
74
9»
71
4*
68
92
74
4a
64
92
69
4*
68
92
66
42
65
9*
II
43
75
93
66
43
70
93
77
43
68
93
70
43
68
93
62
43
68
93
44
64
94
61
44
82
94
61
44
69
94
61
44
65
94
63
44
73
94
79
45
64
95
70
45
65
95
65
45
70
95
70
45
69
95
60
45
74
95
58
46
73
96
64
46
65
96
73
46
66
96
62
46
5a
96
73
46
62
96
70
47
61
97
71
47
69
97
62
47
71
97
73
47
76
97
73
47
63
97
74
48
63
98
67
48
68
98
60
48
75
98
71
48
66
98
77
48
74
98
63
49
67
99
67
49
6+
99
58
'49
62
99
70
49
60
99
7*
49
74
99
73
Total
3450
34»4
3445
3412
34'9
r43o
3362
34»9
3382
3388
§ILj
ENTTMEEATION OP PBIMES IN THE FIEST NINE MILLIONS.
15
TABLE I. {continued).
NujfBEB OF PbIMES IN EACH SUCCESSIVE GBOITP OP 1000 NUMBEBS, FROM TO 9,000,000.
Thibd Million.
2,500,000
2,600,000
2,700,000
2,800,000
2,900,000
to
to
to
to
to
2,600,000
2,700,000
2,800,000
2,900,000
3,000,000
o
h
50
59
63
50
69
7«
50
7«
67
50
67
70
50
69
1
66
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68
I
64
51
72
I
.63
51
67
I
63
51
68
I
55
5*
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1
75
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11
2
67
5a
64
2
62
5a
70
2
71
5a
63
2
66
5a
69
3
66
53
66
3
64
53
67
3
64
53
61
3
74
53
67
3
64
53
59
4
73
54
74
4
67
54
70
4
77
54
59
4
7a
54
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4
70
54
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5
67
55
68
5
67
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5
69
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65
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73
5
70
55
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6
75
56
63
6
63
56
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6
66
56
76
6
69
56
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6
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56
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7
56
57
66
7
60
57
70
7
71
57
61
7
56
57
76
7
73
57
58
8
tt
58
63
8
70
58
75
8
7a
58
68
8
78
58
60
8
68
58
6g
9
66
59
7a
9
68
59
64
9
64
59
72
9
75
59
63
9
70
59
65
lO
67
60
72
10
67
60
64
10
69
60
68
10
71
60
67
10
62
60
76
II
Z3
61
74
II
71
61
68
II
60
61
74
II
73
61
71
II
66
61
67
12
69
62
63
12
69
62
59
12
74
62
73
12
71
62
65
12
64
62
7»
13
66
$3
73
»3
66
63
62
13
68
63
67
13
77
63
67
13
65
63
67
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75
64
62
14
7a
64
81
H
68
64
62
H
63
64
67
14
64
64
65
»5
7a
65
55
15
79
65
74
»5
71
65
7a
15
64
65
75
>5
68
65
77
i6
53
66
75
16
64
66
60
16
66
66
65
16
64
66
7a
16
65
66
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17
61
67
72
17
70
67
63
17
69
67
69
17
75
67
66
>7
68
67
68
i8
68
68
73
18
67
68
67
18
83
68
70
18
65
68
63
18
73
68
69
19
71
69
59
19
56
69
64
«9
67
69
64
19
72
69
59
19
69
69
69
20
63
70
74
20
68
70
5a
20
60
70
60
20
75
70
66
20
71
70
68
%i
65
71
61
21
64
71
71
21
59
71
75
21
68
71
62
21
66
7»
$9
22
74
7»
76
22
66
72
59
22
66
72
61
22
7a
7a
70
22
74
7a
66
13
76
73
62
as
74
73
72
a3
75
73
71
as
66
73
78
as
61
73
67
44
80
74
72
24
7«
74
65
a4
62
74
71
a4
67
74
57
a4
68
74
67
as
67
75
74
a5
63
75
75
a5
59
75
67
a5
66
75
62
a5
7a
75
68
26
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76
7*
26
63
76
71
26
64
76
60
26
59
76
69
26
66
76
65
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58
77
75
a7
68
77
60
a7
61
77
70
a7
56
77
64
a7
58
77
«S
28
65
78
71
28
58
78
63
28
68
78
65
28
64
78
60
28
62
78
61
*9
78
79
62
29
77
79
77
29
71
79
66
29
74
79
65
a9
73
79
7»
30
78
80
73
30
69
80
69
30
66
80
63
30
71
80
70
30
71
80
75
31
67
81
77
31
75
81
65
3»
7»
81
67
31
66
81
65
3»
60
1 81
7>
3»
65
82
65
3a
76
82
62
3a
66
82
69
32
66
82
70
3a
7a
1 82
67
33
58
83
74
33
73
83
71
33
63
83
67
33
83
83
63
33
57
: 83
6l
34
73
84
68
34
65
84
70
34
60
84
75
34
74
84
76
34
67
84
35
64
85
63
35
59
85
70
35
65
85
70
35
65
85
62
35
69
85
73
36
70
86
67
36
67
86
63
36
58
86
64
36
71
86
69
36
74
Z6
7»
37
56
87
68
37
65
87
74
37
65
87
69
37
73
87
68
37
69
87
56
38
80
88
64
38
62
88
65
38
67
88
57
38
69
88
64
38
72
88
64
39
61
89
74
39
70
89
7a
39
75
89
62
39
66
89
71
39
69
89
5«
40
66
90
64
40
66
90
68
40
68
90
60
40
60
90
70
40
62
90
69
41
66
91
67
41
76
9»
68
4»
72
9'
77
41
69
91
61
41
62
9'
67
42
62
9*
62
4a
62
92
67
4a
81
92
68
4a
71
92
63
42
56
92
74
43
67
93
67
43
71
93
67
43
58
93
64
43
71
93
70
43
65
93
69
44
73
94
76
44
67
94
76
U
67
94
68
44
58
94
67
4*
73
94
63
45
68
95
66
45
81
95
64
45
64
95
69
45
63
95
60
45
60 I
95
65
46
64
96
71
46
57
96
67
46
65
96
57
46
67
96
7a
46
7«
96
65
47
59
97
70
47
73
^l
73
47
75
97
67
47
68
97
60
47
58
97
78
48
74
98
64
48
70
98
70
48
62
98
76
48
80
98
64
48
69 ,
98
7a
49
71
99
69
49
66
99
66
49
60
99
78
49
64
99
75
49
73
99
«5
Total
3391
3418
3376
3389
3348
3368
34a7
3319
t
3345
3363
16
INTBODUCnOIK.
[§n.
TABLE I. {continued).
Number op Pbimes in each successitb group op 1000 Numbers, prom to 9,000,000<
Fourth Million.
3,000,000
3,100,000
3,200,000
3,300,000
3,400,000
to
to
to
to
to
3,100,000
3,200,000
3,300,000
3,400,000
3,600,000
62
50
66
5a
50
68
76
50
69
67
50
69
^%
50
74
I
68
51
61
I
66
5»
69
I
69
51
68
I
58
5"
59
I
72
5»
61
%
70
5»
71
2
80
5a
7a
2
74
5a
63
2
65
5a
55
2
7a
5a
55
3
75
53
7a
3
67
53
72
3
71
53
61
3
65
53
67
3
75
53
64
4
65
54
62
4
68
54
66
4
61
54
67
4
70
54
68
4
71
54
7a
5
71
55
62
5
69
55
65
5
58
55
70
5
7»
55
66
5
60
55
67
6
56
56
69
6
67
56
74
6
71
56
63
6
75
56
59
6
63
56
61
7
71
57
63
7
68
57
7X
7
75
57
66
7
57
57
63
7
62
57
69
8
65
58
69
8
71
58
5a
8
68
58
7a
8
73
58
57
8
70
58
66
9
67
59
62
9
68
59
68
9
63
59
63
9
7a
59
60
9
69
59
57
lo
64
60
7*
10
68
60
64
10
74
60
81
10
62
60
70
10
71
60
64
II
76
61
71
11
61
61
59
II
7»
61
63
II
74
61
68
II
74
61
66
1%
71
62
65
12
63
62
62
12
63
62
61
12
64
62
50
12
69
62
60
»3
63
63
60
»3
72
63
73
'3
53
63
68
13
70
63
68
13
63
63
65
«4
70
64
56
»4
64
64
70
14
76
64
61
14
63
64
74
14
7a
64
62
15
60
65
69
15
70
65
73
15
69
65
78
15
67
65
80
»5
58
65
70
i6
61
66
63
16
70
66
79
16
73
66
64
16
7a
66
68
16
65
66
67
17
7«
67
69
17
70
67
64
17
68
67
68
17
57
67
72
17
69
67
64
I8
56
68
69
18
80
68
54
18
69
68
70
18
64
68
65
18
67
68
71
19
67
69
70
"9
59
69
70
19
58
69
63
>9
67
69
73
19
58
69
65
so
75
70
7a
20
68
70
66
20
70
70
67
20
60
70
73
20
74
70
61
%i
65
7»
74
21
67
71
76
21
66
7"
59
21
68
71
59
21
60
71
55
X2
61
72
65
22
65
72
78
22
66
7a
68
22
77
7a
65
22
66
72
67
as
66
73
65
a3
70
73
59
a3
69
73
59
a3
53
73
59
a3
64
73
\l
»4
66
74
79
a4
60
74
69
a4
65
74
7»
a4
77
74
70
24
73
74
66
»5
64
75
68
a5
63
75
60
a5
67
75
76
a5
7a
75
63
a5
66
75
59
26
76
76
73
26
63
76
65
26
66
76
68
26
65
76
71
26
70
76
1!
27
62
77
67
27
76
77
64
27
70
77
75
27
64
77
74
27
7'
77
66
28
74
78
69
28
65
78
73
28
65
78
77
28
70
78
72
28
64
78
66
*9
54
79
63
29
69
79
53
29
68
79
46
29
71
79
61
29
64
79
62
30
7«
80
7*
30
69
80
68
30
63
80
69
30
60
80
66
30
60
80
67
31
64
81
70
3«
7«
81
66
31
68
81
64
3»
62
81
58
31
70
81
65
3»
69
82
60
3*
60
82
74
3a
7a
82
60
3a
60
82
68
3a
68
82
55
33
67
83
63
33
69
83
77
33
71
83
80
33
7a
83
69
33
70
83
67
34
68
84
71
34
64
84
59
34
61
84
62
34
59
84
70
34
64
84
67
35
60
85
65
35
65
85
69
35
7a
85
60
35
67
85
66
35
67
85
57
36
58
86
65
36
66
86
68
36
75
86
70
36
58
86
71
36
77
86
67
37
73
87
. 77
37
63
87
68
37
66
87
69
37
74
87
75
37
72
87
63
38
67
88
72
38
65
88
68
38
75
88
68
38
67
88
62
38
65
88
63
39
60
89
71
39
74
89
74
39
64
89
68
39
66
89
64
39
7a
89
65
40
7«
90
63
40
57
90
61
40
64
90
72
40
74
90
70
40
62
90
63
41
69
9«
76
4«
65
9^
69
41
61
91
70
41
70
91
67
41
73
9»
li
4*
66
92
67
4a
73
92
69
4a
62
9a
61
42
74
9a
53
4a
64
92
68
43
Z^
93
68
43
56
93
67
43
64
93
60
43
61
93
67
43
71
93
67
44
64
94
58
44
7»
94
68
44
66
94
67
44
64
94
70
44
75
94
72
45
66
95
68
45
67
95
63
45
66
95
61
45
63
95
70
45
69
95
60
46
63
96
69
46
65
96
74
46
70
96
66
46
66
96
66
46
53
96
67
47
7a
97
62
47
66
97
66
47
70
97
75
47
65
97
76
47
64
97
56
48
60
98
66
48
73
98
71
48
61
98
63
48
68
98
1^
48
70
98
68
49
65
99
57
49
65
99
67
49
65
99
53
49
70
99
65
49
69
99
7a
Total
3320
3356
3343
3374
3368
33a3
3330
3309
_
3370
3241
^n.]
ENUMEEATION OF PSIMJS IN THE FIEST NINE MILLIONS.
17
TABLE I. (continued).
• • • • ' ' .
Number op Peimes in each successive group of 1000 Numbers, from to 9,000,000,
Fourth Million,.
•
•
1
•
a
i^mm
•
3,600,000
•
•
• 3,600»000
3,700,600
•
3,800,00C
) •
.3,900,000 j
t
to
• 1
•
to .
to
to.
to
•
3,000,000
,
' 3,700,000
•
•
•
•
3,800,000
•
3,900,000
•
4,000,000
- •
•
o
56 .
SO
^4
73'
50
^8
67.
50
. ^0
•
61
SO
•
57
57
SO
66
I
59-
s»
6s .
J
•67
51
49
I
67
51
69
I
73
SI
64
I
69
SI
. 69
z
63
.52
65
1
66
5a
67
2
65
Sa
69
2
. 61
sa
66
. 2
55
sa
58
3.
83
53
62
3'
•58
53
59
3
67
S3
7a
3
^■•
53
64
3
67
S3
60
•
4
66
54
^4
4
73
54
64
4
67
•
54
62
4
80
54
67
4
62
54
60
5
73 •
55
73
5
•
71
55
73
5
' 66
55
59
5
64
55
63
5
60
55
.73
6
53
56
74
6
68
56
72
6
.55
56
74
6
63
56
71
• 6
69
56
61
7
' 66
S7
62
7
77
57
71
7
70
57
65
7
. 61 •
S7
65
7
77
57
68
8
• 63
58
67
8
.63
58
64
8
7a
58
65
8.
55
58
66
8
66*
58
61
9
69
59
7* .
9
•76
59
63
9
59
5?
62
9
74 '
59
67
9
69
59
67
lO
«i
60
62
, 10
66
60
63
10
54
60
57
10
76
60
65
10
• 74
60
63
II
J4
61
62
II
61
61
59
II
71
61
S^
11
62
61
64 .
II
•55
61
64.
■12
73
62
• 68
12
79
62
73
12
74
62
. 57
12
69
62
70
.la
60 .
62
.61
>3
69 .
^3
68.
13
69
63
66
»3
66
63
68
13
60
63.
61
13-
55
63*
60
'4
64
64
• 68.
H
7a
64
67
H
68
64
.60
H
76
64
68
14
71 .
*
64
76
;5
65
65
•
71
1-5
58
65
69:
»5
64
65
>9.
is
7a
65
•64
»S
70
6S
•
5?.
16
59
66
6a
16
68
66
67
16
63
66
64
16
66
66
58
16
62
66
59*
17
60
67
76
17
73
•67
61
17
7a
67
74
17
72
67
.69
17
.56
67
68
i8
65
'68
63
18
64
68
68
18
66
68
63
18
67
68
69
18
'66
68
65 •
li
73
69
. 70
«9
60
69*
69
19
73
69
64
«9
62
69
58
19
79
69
71
zo
66
70
69
20
66
76,
68
20
64
70
62
20
7J
70
• ^7
20
65
70
66
21
62
7»
65
21
. V
71-
63
21
67
7^
60
21
71
71
61
21
59
71 •
66
22
63
7a
56
22
8a
7a
78
22
63
72
^7
22
70
7a
62
22
70
7a
63.
»3
58.
73
66
23
. 62
73
65
i3
62
73
73
a3
n.
73
70
a3
64
73
70
44
70
•
74
64
24
71
74
• M
24.
78
74
71
a4
74
68
a4
75
74
74
as
59-
75
66
•
as
62
•75
67
as
65
•75
67.
as
. 76
75
65
as
• 6s
75
58
26
57
76
67
26
71
76
• 75
26
80
76
60
26
4^
76
62
26
64
76
74
27
65
77-
. 62
a;
63
77
7a
27
63.
77
77
a-7
64
77
.66
27
71
77
78
28-
71
78
68
28
67
78
79
28
67
78
•61
28
70
78
61
28
68
78
60
29
59
79
57
>9
71
79
71
29
78
•
79
67
29
76
79
69
29
74
79
59
30 .
65
80
67
30
7a
80
s«'
• •
30.
.69
80
70
30-
71
80
67*
.30
67
80
65
3»
72
81
70
.3»
61
81
60
31
69
81
58
31
58
81
66
31
63
81
73
32 .
V'
82.
..65
32
68
82
64
3a
■6s
82
V
32
66
82
64 .
32
67
82
SS
33
67
83
67
33
7t
83
62
33
•62
83
66
33
P
^3
70
33
61
83
H
34
62
84
67
34
61
84
68
34
59
84
73
34
60
84
55
34
61
84
9
5^
35
79
85
62
•
35
60 •
85
63
35
7a-
85
.65
35
• 7a
85
•69
35
64
85
60
36
61
86
78
36
62'
86
71
36
68
86
72
36
60
86
73
36
61
86
58
37
66
87
71
37
62
87
. 62
37
69
87
66
37
•7*
87
7a
37
78
87
38
•53
88
67
•38
66
88
73
38
60
88
57
38*
62
88
56-
38
66
88
64
39
• 66
89
53
,39
71
89
59
39-
•65
89
67-
39
66
89
69
3'9
^0
89
66
40
59
90
74
40
64
90
. 7a
40
58
90
60
40.
• S^
90
71
40
78
90
60
4»
70
9«
t
4>
74
91
68-
41
66
9^
67
41
81
9'
69
41
11
91
. 63
4*
64
92
4a
1*
92
61
4a
•^J
92
72
4a
67
92
67
42
68
92
75
43
64
93
6s.
43
67
93
64
43
66
93
• 57
43
61
93
75
.43
76
93,
68
44
59
94
70
4*
56.
94
7*
44
61
94
59
44
65
94
73
44
68
94
6t
45
71
1
95
67*
45
70
:95
7a
■45
65
95
63-
:i
68
95
63
45
55
95,
66 .
46
70
96.
. *7
46
. 69
' 96
• z*
46
68
96
71
69
9d
68
46
73
96.
54
47
75
97
•77
4?
59
1 97
60
47
56
97
66
47.
60
97
61
47
69
97
61
48
• 61
•98
66
48
61
;»«
68
48
81
98
^5
.48
75
98
!5
48
67
98
70
• 49
61
99
• 6S
49
60
|.99.
1
70
•
49
65
•
99
67
•
49
66
99
69
•
49
69
•
99
63
Tot«l
3261
3314
•
'
3338 .
(
3333
3320
•
327Q
3335
«
1
•
3289
3310
•
•
322s
• /
18
INTBODFCnON.
[§II.
TABLE 1. (continued). '
Number of Fbimss in each suogessivb gboup of 1000 Numbebs, from to 9,000^000.
TiFTH Million.
,
4,000,000
•
•
4,100,000
•
4,200,000
«
•
•
4,3OO'ob0
•
r
•
4,400,000
•
to
•
to
to
•
•
to
. t
to •
•
•
4,100,000
•
4,200,000
\
4,300,000
•
4,400,000
i
' 4,500^000
•
•
O
68-
50
68
•
6^
50
•
70
71
50
7»
7a •
50
67
«5
50
66
I
69
51
58
I
60
5»
«4
I
65
51
63
I
58
51
60
I
60
51
65
1
7'
5*
67
2
66
5a
64
2
68
5*
P
. a
70
5»
64
2
l^
5a
61
.3
|9
53
68
3
60
53
65
3
63
53
69
3
79
53
P
3
«4
53
7a
•
4 •
«♦
54
S«
4
69
54
64
4
63
54
64-
4
«3
54
61
4
6i
54
65
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63
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86
70
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76
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62
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37
67
57
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37
66
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70
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37
70
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64
38
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70
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63
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64
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67
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58
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63
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68
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43
67
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68
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75
93'
60
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69
93
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43
61
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62
43
67
93
61
44
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. 66
44
64
94
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67
94
65
44
63
94
60
44
62
94
73
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•
45
64
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66
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64
95
68
:i
55
95
63
45
63
95
62
45
* 68
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' 59
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45
• 77
67
46
53
96
66
61
96
64
46
62
96
73
46
64
96
. 69
•
47
62
57
63
47
69*
98
60
47
63
97
59
47
68 .
97.
. 57
47
66 .
97
6a
i
70
98
75
48
75
60
48
7*
98
74
48
67
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62
48
59
98
64
62
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. 58
49
68
99
66
49
69
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49
60
99
67
49
76
99
. 6a
•
Total
«
3344
•
3284
3236
.3304
•
•
3269
•
3H»
3260
3251
•
•
3331
3a8»
tiL]
bnuMeeation op petmes m the fiest nine millions.
19
TABLE I. (cmiUnued).
• • • . *
•NUMBBE OP PbIMBS IN BACH SUCCESSIVE GROUP OP 1000 NUMBBBS, PBOM TO 9,000^000.
. . • ^
. FiPTH Million.
m
4,500,000
.
4,600,000
4,700,000
»
1
•
4,800,00C
\
•
4,900,000
• to
to
to
to'
to
•
4,600,000
•
«
»
4,700,000
1
•
4,800,000
1 ••
4,900,00C
> .
6,000,000 ..
t
o
68
50
68
67
50
58
76
50
67
65
50
67
•
6
75
50
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72
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70
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3
69
53
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53
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56
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63
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4
59
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55
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5
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60
55
70
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56
69
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56
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70
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56
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71
7
62
57
63
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57
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7
68
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74
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58
60
8
68
58
63
8
67
58
72
8
66
58
71
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• 73
58
51
9
66
59
62
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1
9
67
59
59
9
67
59
65
9
58
59
61
9
•
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59
54
10
60
60
58
10
70
60
70
10
59
60
66
10
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75
10
58
60
74
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64
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64.
61
64
11
63
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60
61
62
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69
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61
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72
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65
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63
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73
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67
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77
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64
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66
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69
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. 63
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60
85
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35
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85
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63
85
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35
84
85
64
35
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64
16
65
86
60
36
64
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36
67
86
68
36
67
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69
36
63
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77
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58
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68
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61
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74
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70
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66
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. 76
87
66
37
58
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67
38
65
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59
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66
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63
38
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65
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68
88
70
38
70
88
62
39
66
89
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70
89
59
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53
89
66
39
59
89
64
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68
89
79
40
66
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67
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66
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60
40
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40
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77.
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67
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61
91
65
41
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92
67
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59
92
63
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73
92
63
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63
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54
92
64
43
66
93
63
43
66
93
66
43
64
93
58
43
58
93
78
43
69
93
66
44.
60
94.
76
44
56
94
65
44
59
94
68
44
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94
63
44
76
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75
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66
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67.
45
65
95
67
46
70
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70
46
71 •
96
75
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65
96
68
46
65
96
69
46
7a
96
57
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61
97
66
47
73
97
69
47
• 66
97
65
47
70
97
68
47
58
97
57
48
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98
66
^
59
98
61
48
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98
65
48
67
98
• 6t
48
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98
67
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61
99.
*
54
49
73
99
55
•
49
• •
• 62
•
99
66
49
6j
99
72
49
69
•99
71
Total
«
3216
3277
3265
•3258
3a55'
3220 '
3*56
•
3298
•
3a43
3a79
d2
• •
.20
IKTBODFCnON.
[§II.
TABLE I. (continued).
Number op Primes in bach successive group op 1000 Numbers, prom to 9,000,000,
Sixth Million,
•
•
6,000,000
■•
6,ibo,boc
•
•
«
5,200,00c
\
j5,800,00C
L
•
5,400,000
•
* to
to
'
to
to
to .
5,100,000
%
•
5,200,000
•
r
•
•
5,300,000
, »
.
5,400,000
)
• •
6,500,009
o
63
50
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64
50
•
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5.0
68
69
50
56
i
61
50
80
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69
51
73
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55
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58
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67
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67
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70
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71
53
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3
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3
67
53
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3
66
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54
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4
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4
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54
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59
54
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55
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5
48
55
49
5
68
55
63
5
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55
68
5
• 63
55
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6
71
56
71
6
66
56
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6
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56
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6
67
56
65
6
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7
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7
68
57
58
7
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57
59
7
61
57
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8
58
58
69
8
66
58
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59
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8
63
58
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8
70
58.
72
9
68
59
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59
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59
64
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60
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10
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63
60
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61
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55
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63
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10
70
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86
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58
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37
69
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38
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39
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. 66
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55
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45
64
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46
60
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40
6a
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70
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67
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63
46
6b
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68
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97
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67
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58
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63
49
70
99
60
49
77
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61
49
63
99
66
49
61
99
72
Total
•
3ao&
3a56
»
3aft7
3209*
3278
«
3*15
3231 .
3231
3axa
3aa6
i
§•".]
ENTJMEEATION OF PEIMBS. IN THE FIEST NINE MlLLIOlfS.
21
TABLE I. (continued).
NuMBEtl OP PBI3ifiS IN BACH SUCCESSIVE GEOUP OP 1000 NuMBEBS, PEOM.O TO 9,000,000,
Sixth Million.
•
5,500,000
•
•
•
6,600,00C
)
t
5,700,00C
1 '
5^800,000
f
5,900,000
«
to
t6
•
•
to
•
•
to
«
to
•
5,600,000
•
5,700,000
•
<
5,800,000
*
5,900,000
6,000,000
•
o
58
50
62 '
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• 69
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71
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•
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40
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63
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•
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40
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71
41
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• 50.
43
57
93
72
43
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43
.69
93
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43
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44
64
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44
69
94
57.
44
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44
63
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65
44
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94
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45
62
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45
63
95
. .54
45
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95
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45
56
•95
60
46
60
96
66
46
57
96
p
46
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96
61
46
77,
96
54
46
61
96
57-
47
60
97
61
47
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97
. ^5
47
66
97
67
47
75
97
67
47
68
97
• i9
48
71
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67
48
60
98
73
48
72
98
71
48
62
98
57
48
64
98
71
•
.49
61
•
99
70
•
49
61
99
66
49
68 .
99
61
49
62
99
69
49
. 58
•
99
68
Total
3179
3223
3176
3228
•3210
3167.;
321 1
•
3225
4
3H3
3«77
22
INTE0DT7CTI0N.
[§■«.
TABDE I. (continued).
NUMBBE OP PeIMES IN BACH SUCCESSIVE 0BOUP OP 1000 NUMBBES, PEOM TO 9,000,00Q.
Seventh Million.
*
•
•
6,000,00Q
•
1
•
6,100,000
1 * •
6,200,000
)
6,300,0(K1
1
. 6,40O,00C
)
to
•
•
to
•
to
f
to
to
•
•
6,100,000
•
6,200,00c
•
1
•
•
6,300,00C
•
•
6,400,000
•
•
• .
6,600,00C
•
1
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66
50
68 ^
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65
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34
60
84
66.
35:
59
85
. "50
35
66
85
«3
35
67
85
63
35
66
85
60
35
63
85
56
36
F
86
70
36
62
86
62
36
62
86
62
36
•«4
86
. 57
36
S8
86
• 66
37
66*
\L
66
37
66
87
65
37
54
87
• 72
37
65
87
• 59
37
«5
.^7
• 53
38
60
%%
77
38
6s
88
56
38
69
88
66
38
67
88
• 59.
38
63
'88
66
39
.67
89
61 •
39
69
89
70
•39
59
89
59
39
54
89
71
39
.60
89
67
•
40
68
90
67
40
66
90
64
40
66
90
59
40
%
90
61
40
70
90
7»
4«
63
91
62.
41
61
91
55
4»
. 74
9»
71
41*
9»
61
41
65
91
60
4»
69
92
68
4*
.62
9»
55
4*
55"
«»
56
4»
69
9*
60
4»
67
9»
60
43
63
93
55
43
61
•
93
67
43
65
93 ,
61
43
66
93
^?
43
58-
93
59
44-
61
94
?^
44
63
94
70
4*
58
94
74
44
61
94
66
44
70
94
69
*!•
63
95
70 •
•45
63
95
' • 62
45
68
95
65
45
61
95
67
45
59
95
60, <
46
56
96
63
46
60
96
73
46
60
9«
73
4^
64
96
57
46
60
96
65
47
62
97
63
47
62
97
66
47
66
97
65
47
• 48
^7
69
47
63
97
62
48
58
98
' 7»
48
67
98
7»
48
58
98
76
48
70
98
7«
48
62
^^.
70
49
64
99
66
49
72
.99
66
49
67
99
60
49
69
99.
64
49
59
99
62
1 (Total
319'
3206
•
3203
3199 *
3168
3*57
3192
3«45
•
3152
3195
§n.]
BIJUMERATION OF PBIMBS JN THE FIEST NINE MILLIONS.
23
TABLE I. (continued).
NuMBBB OF Primes in BAcfi sttccbssivb group, of 1000 Numbers, from to 9,000,000,
Seveoth Million.
6,600,000
•
6,600,000
«
6,700,000
6,800,000 •
. 6,900,000
• to
to
•
to
to
•
•
to
%
6,600,000
•
•
" 6>700,000 .
6,800,000 .
«
•
6,900,000
• 7*000,000
o
74,
50
60
60
•
50
58
9.
.63
50
64
62
50
67
61
50
60
X
64
51
56
I
68
5"
60
I
66
51
70
I
58
fi
61
I
60
51
75
2 '
• 74
Sa-
71
2
53
5»
75
2
6s
5*
•.60
2
•7.7
52
70
2
. ^5
5»
70
. 3
63
53
•58
3
65
53
56
3
64
53
62
3
59
53
f9
3
50
53
76
4
62
•
54
68
4
•
.63
54"
«3
4
59
54
68
4
64
^
66
4
64
54
55
5
S8
55
63
5
71
55
65
5
63
55
. 67
5
• ^5
55
66 .
1
68*
55
.67
6
55
56
58
6
.67
56
6%
6
7»
56^
57
6
66
56
65
68 .
56
56 .
7
67-
S7
60
7
57
57
61
7
71
57
66
7
68
57
. 69
7
61
57
56
S
«S
58
7»
8
66
58
54
8
^
58
66
8.
56
58
59
8
66
58
57
• 9
«o •
59
74
9
61
59
59
9
65
59
7t^
•
9
57
59
. 7'
9
68
59
59
•fo
• "67.
60
68
10
72
60
67
10
61
60.
68
10
69
•60
61
10
57
60
70
II
61
61
70
II
IV
61
66
II
69
61
53
II-
77
61
67
II
68
61
65
12
76
62
64
12
<S2
6g
12
• 56
62
56
12
57
62
It
12
56
62
63 .
13
61
63
57
»3
67
63
61
13
66
63
•
7«-
'3
61
63
13
69
f3.
67
'4
57
64
59
'4^
6s
64
65
14
69
64
61
H
69
64
65
H
56.
64
79
IS
S<
55
.64
15
.69
65
«5
15
. 75.
65
63
»5
62
H
6s
»5
60
65
59
i6
75
66
65
16
.69
66
69
16
56
66v
57
16
60
66
70
16
62
66
62
17
69
67
59
17
65
67
17
6S
67
65
17
66
67
63
»7
68.
67
64.
i8
60
68
7<5
18
69
68
65
18
60
68
18
71 •
68
69
18
71
68
63
19
S8
69
58
19
59
69
6s
19
57
•
69
•
66
»9
61
69
60
•
»9
7A-
6-9
56
20
70
70
• ^
20
61
70
il
20
63
70*
.^3
2Q
69
70
55
20
1*
70
58
21
59
71
63
21
58
71
21
77
71
62
21
70
71
58
21
6b
71
7«
22
63
7»
.67
22
•58
72
60
22
65
72
62
22
53
72
62
22
67
72
57
»3
60.
73
63
23
• 61
73
66.
»3
63
73
68
*
a3
60
•73
70
23
58
73
7»
*4
7»
•
74
61
»4
59
74
68
24
65
74
60
24
58
74
68
24
59
74
66
*5
64
75
•«7.
»5
67
75
62
25
70
75
^11
25
68
75
59
25
66
75
63
26
57
76
65
26
- 61
76
66
26
54 .
76
26
^3
76
.70
26
61
76
63
27
56
7^7
$J
27
«♦
77
66
27
69
77
61
27
68
77
55
27
f^
77
28
. 75
78
63
28
62
78
1^
28
57
78
70
28
^ ^3
78
74
28
63
78
60 '
29
70
79
60
»9
64
79
»9
63
79
64
*9
62
79
5»
29
67
79
66
. 30
63
80
66
30
68
80
' 60
*
30
64
80
«3
30
60
80
66 .
30
65
80
63
.3«
.65
Si
70
31
60
81
67
3>
64
81
59
3.1
65.-
81
68
31
59
81
61
3»
62
82
70
32
57*
82
60
32.
67
82
65"
32
6^
82
70
3»
64
82
*2
I5
33
61.
83
73
33
63
83
. 1?
33
63
83
65
33
• 64
83
59
33
69 '
V
34
5^
84
66
34
62
84
34
65
84
63
34
60
84
73
34
69
84
35
61
85
' 6,
35
64
85
65
35
66
85
61
35
• 57
85.
63
35
66
85
60
36
70
86
' 66
36
55 .
86
60
36
64.
86
64
36
64
86
65
36
69
86
64
37
1J
«7.
64
37
7»
87
S6
37
64
87
61
37
69
87
53
37
61
.87
69
38
88
58
38
• 55
88
67
38
59
88
68
38
62
88
66
38
75
88
61
39
6i
89
64
39
64
89
63
.39
58
89
5«
39
61
89
68
39
55
89
68
40
59
90
54
40
73
90
it
40
66
90
60
40
60
90
62
40
1
59
90
65.
4«
68
91
70
41
68
91
41
63
91
5«
41
60
91
68
41
68
91
.60
4»
66
92
■6z
4*
61
94
11
4*
53
9*
6a
4»
79
92
66
42
63
92
58
43
68
93
63
43
66
93
66
43
64
9>
.^
43
55
93
• '54
43
61
93
59
4* •
55
94
60
•
44
59
94
60
44
68
94
.6.
44
•
67
94
. 70
H
60
94
72
45
73
95
63
45
70
95
63
45
70
95^
65
45
61
95
72
45
62
95
66
46
62
96
• 66
46
60
96
64
46
63
96
55
.46
70
• 96.
64
46
62
96,
57
•47
68
97
58
47
. 71
97
. 71.
47
70
97
70
47
55
97
52
47
75
97
70
4«
64
98
7*
48
55
98
6a
48
62
98
69
48
60
98.
64
48
.66
98
68
49
58
99
64
49
57
99
63
49
62
99
65
49
s€m
99
59
49
• 66
99
62
Total
3198
3204
•
3178
3160
3210
>
3165
•
3183
3228
3187 .
3178
24
INTEODUOTION. '
[§U.
. TABLE I. (continued).
»
NUMBBB OP PbIMBS IN BACH SITCCESSIVB GBOUP OF 1000 NUMBBBS, FROM TO 9,000,000. * .
Eighth Million.
9
7,0.00,090
•
7,100,00c
»■
7,200,000
•
7,300,000
7,400,000
»
' to
•
•
to
*
'to
. to
to
7,100,000
•
« *
7,200,00c
•
9
7,300,000
•
. 7,400,000
•
7,600,0.00
m
o
S5
SO-
62
P
80.
50
60
.
4
70
*
.5©
62
il
50
. 58 •
6S
50
.6*
I
63.
51
66
I
. ^0
51
70
I
63
51
64
I
51
. ^7
I
67
51
6a
2
66
5*.
61 •
2
7»
.Sa
65
'2-
S8
sa
IZ
2
. «o
sa
69
2
1' '
•sa
65
3
66
53
55
3
68
53
61
3
• 71
53
66
3
!5
53
57
1
<5 1
53
57
4
59
54
72 •
4
<7
54
58
4
<S
54
6a
4
64-
54
7a
4
60 '
54
59
5
64/
55
• 60
5
*
57
.55
5?
5
5«
55
64
5
^
55
65
5
57
•
55
61
6
63
56
66
6
67
56
71
6
60
56
11
6
6a
56
69
6
62
56
^5
7
66
57
61
7
62
57
64
7'
67
57
7
if
57
61
7
6)
57
6a
8
61
5«
64
8
61
58
71
8
60
58
57
2
h
58
65
8
58 ;
58
59
9
^6
59
61
9
69
sr
63
9
62
1
61
9
58
59
59
9
. 65
• 1
59
6y
■lO
63
60
61
10
60
60
•
57
10
«4
60
71
10
62
60
63
10
66
1
6o'
65
II
50
61
63
II
62
61
6j
II
. 66
61
76
•II
54
61
74
II
58
61
48
i\
1?
62
. 50
12
59
62
.58
12
6a
6a
59
12
64
62
55
12
63
6i'
63
'3
63
60
13
66
63
63
13
65
63
73
13"
..g
63
63
13
68
63
67
H
69*
64
; 63
14
68
64
.66
14
59
64
62
14
64
62 •
14
73
64
61 .
is
55
^5
68
15^
68
H
«3
15
6a
'5l
64
IS
64
65
61
15
71
65
60
i6
70
66
53
16*
56
66
7»
16
70
66
5«
16
79
66
73
16
60'
66
57
17
.69
67
61
17
S8
67
56
17
58
67
70
*7
63
57
60
.17
67
.56
i8
70
68
62
18
. 56
68
73
18
7«
68
59
18
il
68
65
18
60
68
• 67
»9
59
69
53
■19
56
69
59
19
53
69
61
> 19
^9
67
19
67.
69
66
20 <
7»
. 70
75
20
69
70
67
20'
64
.70
65
26
68
70
55
20
7a
70
. .h
21
56
71
70
21
59
71
67
21
67
71
63
21
63
71
62
21
54
71
57
22
66
72
67
22
67
•72.
II
22
«4
7a
69
22
61 .
72
55 .'
2>
^3
7a
it
.»3
65 -
.73
60
a3
64
73
43-
67
73
63
23
61
73
73
a3
67
73
!24
0a
74
68
•
24
69
74
68
H
64
74
59
a4
7a
74
73
a4
. 57
74
75 •
as
63
75
63
as
60
75
54
11
67
75
Z^
:i
60 .
li
61
as
•70
75
74
26
6$
76
<$
26
54
76
43
64
76
61
60
' .67
26
70
76
66
aZ .
64
77
. «»
a7
*59
77
. 64
11
55"
77
. ^7
27
tl
77
61
a7
c6
77
67
28
79
78
«s
28
56
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70.
. 67
78
60
«
28
78
7a •
28
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78
62
29
61
79
60
29
54
79
62
29
57
79
62
a9
61
79.
56
29
59-
79
65
30
64-
5°
S9
30
64
80
66
30
5a
80
60
30
58
80
73
30
68
80
60
3« .
66
81
64
3«
61
81
67
3''
68
81
56
31
63
81
57
31
7a
81
62.
3»
63
82
64
3a
•68
82
55
3a
61
82
7/
32
62'
82
66
32
59
82
64
33
63
H
6a.
33
62
83
• 60
33
69
83
66
33
.62
83
69
33
64
83'
65
34
• 70
84
66*
•
34
58
84
. 69
34
68
84
55
34
66
84
61
34
60
84
62
*
.3S
65
85
60
35
.68
35
64
35
56
85
66
35
77
85
63
35
t
71
85
64
36
62
86
67
36
65
86
65
36
58
86
57
36
55
86
53
36
54
86
64
37
6i«
87
62
37
63
87
60 ,
37
• 7a
87
66
37*
BS
87
55
37
57
a7
68
38
67
88
61
38
61
88
64
38
60
88
74
38
•53
88
69
38
66
88
60
39
67
89
75
39
47
89
60
39
61
89
65
39
65
89
6ft
39
•
65-
89
•
63
40
71
90
65
40
77
90
66
40
7P
90
62
4»
62
'9a
65
40
58 •
90
58
41-
67
9»
73
4»
70
9«
69
41
62
91
67
41
58
1 91
61
41
73
91
68
•42
61
9a
*7
4a
61
92
.63
4a
• 66
92
68
4a
70
• 9a
S3
42
60
92
5'
.41
60 •
93
'67
43
7^
93
53
43
•67
93
66
43.
S3
. 93
64
43
64 •
93
65
44
69
94
60
44
S3
94
63
4f
67
•
94
sa
4f
58
; 94
61
44
77
94
s*
4S
68
95
58
45
68
95
62
45
• (5i
95
65
45
65.
95
63
45
60
95
58
46
65
9.6
. 68
46
59 .
96
66
46
62
96
. 75
46
61
96
68
46
58.
96
6a
47
72 .
97
.61
47
70
^l
65
47
68
97
60
47
59
97
60
■47
73
97
64
48
|5
98
73.
48
S3
^98.
61
48
59
98,
60
48
.77
98.
60
48
68
98
. 68
49
•
65
99
.57
49
7a
^99
70
49
7«
99
•
57
49
58
99
64
49
65
99
63
Total
3218
3151
•
3»S4
3152
3176
3172
•
3134
3167*
3186
3115
§n.]
ENmiEBATION OF FBIMES IN THE FIRST NINE MILLIONS.
26
TABLE I. {cmtinmd).
NUMBBE OP PbIMBS IN BACH SUCCBSSIVB GEOUP OP 1000 NUMBEES, *PEOM TO 9,000,000,
Eighth Million.
7,500,000
7,600,00C
\
7,700,00c
)
7,800,000
7,900,000
to
to
to
to
to
7,600,000
7,700,00C
)
7,800,000
\
7,900,000
8,000,000
o
62
50
f3
69
50
60
61
50
56
62
50
62'
56
50
7«
I
60
51
67
I
62
5»
62
I
67
5'
63
I
70
51
69
I
59
5«
55
2
65
5»
H
2
67
5»
63
2
58
5*
81
2
5*
5»
65
2
57
5*
60
3
§9
53
1*
3
60
53
63
3
60
53
57
3
62
53
59
3
5*
53
H
4
67
54
60
4
68
54
68
4
62
54
62
4-
70
54
70
4
64
54
51
5
7»
55
63
5
70
55
6c
5
59
55
66
5
7*
55
64
5
66
55
64
6
55
56
67
6
64
56
58
6
58
56
54
6
61
56
54
6
7»
56
57
7
62
57
70
7
68
57
60
7
55
57
70
7
62
57
64
7
66
57
61
8
66
58
55
8
69
58
68
8
69
58
59
8
66
58
68
8
65
58
58
9
58
59
73
9
60
59
65
9
63
59
58
9
61
59
57
9
73
59
63
lO .
5»
60
62
10
65
60
65
10
61
60
61
10
65
60
67
10
56
60
71
II
64
61
53
II
62
61
63
II
76
61
68
II
60
61
67
II
59
61
64
12
59
62
70
12
68
62
53
12
64
62
64
12
73
62
63
12
66
62
63
13
75
63
65
13
53
63
66
13
54
63
65
«3
76
63
69
13
65
63
64
14
58
64
5*
14
57
64
64
14
70
64
64
'4
64
64
63
14
67
64
64
«S
67
65
58-
15
58
65
62
«5
60
65
1?
»5
53
65
53
15
57
65
60
i6
68
66
55
16
64
66
65
16
67
66
16
65
66
7"
16
59
66
61
17
73
67
60
17
66
67
65
17
62
67
60
X7
59
67
54
17
69
67
59
i8
5»
68
61
18
62
68
60
18
66
68
60
18
54
68
65
18
56
68
61
19
59
69
79
»9
62
69
63
»9
65
69
49
19
64
69
62
19
62
69
67
20
62
70
65
20
5f
70
65
20
69
70
7«
20
61
70
«4
20
61
70
«5
21
66
71
66
21
56
71
65
21
55
71
68
21
72
7«
74
21
63
71
58
22
1^
7»
54
22
55
7»
70
22
70
7*
65
22
60
7*
S7
22
\^
7»
68
a3
61
73
61
*3
70
73
64
»3
72
73
73
»3
71
73
66
a3
65
73
!5
*4
64
74
57
H
79
74
60
H
56
74
71
^4
62
74
67
a4
61
74
60
»5
55
75
56
*5
66
75
59
*5
56
75
55
as
60
75
60
a5
62
75
66
26
56
76
63
26
70
76
60
26
60
76
53
26
• 50
76
60
26
66
76
70
»7
56
77
62
27
68
77
. 64
27
62
77
58
27
62
77
6a
27
65
77
68
28
63
78
65
28
68
78
64
28
59
78
69
28
62
78
7»
28
64
78
63
29
7»
79
59
»9
61
79
59
29
71
79
64
29
64
79
65
29
64
79
55
30
73
80
69
30
68
80
61
30
62
80
70
30
62
80
68
30
57
80
60
31
65
81
65
3«
53
81
62
31
55
81
67
31
67
81
62
3»
62
81
62
3»
59
82
5»
3»
56
82
73
3»
64
82
60
3a
57
82
66
3*
74
82
63
33
67
83
66
33
60
83
56
33
58
83
65
33
59
83
68
33
57
83
72
34
63
84
62
34
67 •
84
65
34
64
84
58
34
66
84
61
34
70
84
60
35
62
85
57
35
61
85
69
35
7»
85
55
35
68
85
58
35
59
85
66
36
63
86
61
36
65
86
59
36
5«
86
5»
36
.58
86
60
36
1*
86
69
37
65
87
77
37
57
87
69
37
53
87
64
37
68
87
64
37
66
87
67
38
54
88
63
38
64
88
64
38
70
88
58
38
64
88
70
38
57
88
64
39
67
89
65
39
60
89
62
39
58
89
63
39
73
89
64
39
69
89
51
40
69
90
62
40
58
90
68
40
66
90
66
40
61
90
!3
40
61
90
56
41
Z*
91
59
41
70
91
58
4'
60
91
58
41
ii
91
63
4»
67
91
70
4»
60
92
65
42
58
9»
70
4»
60
92
56
4*
68
92
66
4»
59
9*
5»
43
62
93
74
43
68
93
65
43
65
93
67
43
68
93
58
43
69
93
67
44
62
94
63
44
65
94
59
44
65
94
65
44
69
94
63
44
62
94
62
45
66
95
62
45
56
95
71
45
53
95
63
45
68
95
65
45
73
95
64
46
65
96
66
46
68
96
61
46
62
96
69
46.
64
96
64
46
64
96
63
47
64
97
63
47
66
97
64
47
58
97
62
47
71
97
56
47
64
97
5t
48
69
98
64
48
70
98
61
48
73
98
66
48
60
98
65
48
66
98
66
49
62
99
66
49
63
99
66
49
60
99
61
49
53 '
99
63
49
55
99
5»
Total
3156
3«49
3176
3171
3116
3129
3184
3180
3H9
3125
B
26
INTEODTJCnON.
[§IL
TABLE I. (continued).
Number of Fbimes in each successive gboup of 1000 Numbbbs, from to 9,000,000.
NmTH Million.
8,000,000
8,100,00c
)
8,200,000
1
8,300,00c
\
8,400,000
to
to
to
to
to
8,100,000
8,200,000
1
8,300,000
1
8,400,000
8,500,000
o
76
50
60
57
50
70
*
54
50
67
74
50
56
69
50
62
I
53
51
57
I
67
51
7»
I '
70
51
64
1
63
51
73
I
71
51
65
2
59
5*
67
2
53
5*
57
2
61
5»
58
2
62
5a
65
2
59
5a
64
3
64
53
75
3
67
53
65
3
61
53
62
3
68
53
57
3
54
53
68
4
66
54
58
4
55
54
66
4
58
'54
58
4
60
54
65
4
55
54
61
5
61
55
65
5
62
55
65
5
71
55
64
5
60
55
54
5
62
55
«5
6
62
56
80
6
64
56
61
6
75
56
56
6
64
56
65
6
57
56
63
7
66
57
66
7
67
57
71
7
62
57
62
7
56
57
61
7
68
57
63
8
66
58
61
8
58
58
60
8
62
58
70
8
65
58
67
8
69
58
73
9
64
59
63
9
69
59
61
9
65
59
57
9
59
59
64
9
53
59
57
lO
58
60
61
10
' 63
60
65
10
66
60
59
10
69
60
5*
10
63
60
60
II
59
61
65
Ji
63
61
62
II
68
61
56
II
61
61
58
It
66
61
66
12
67
62
72
12
71
62
71
12
60
62
7*
12
73
62
57
12
68
62
64
13
57
63
64
»3
56
63
64
13
64
53
57
13
64
63
59
13
64
63
1^
14
60
64
58
14
54
64
65
14
67
64
57
14
60
64
62
14
61
64
67
15
64
65
67
15
65
65
63
15
59
65
61
15
60
65
65
15
71
65
61
i6
62
66
54
16
70
66
59
16
66
66
62
16
61
66
59
16
56
66
58
17
63
67
71
17
66
67
66
»7
64
67
58
17
54
67
62
17
56
67
54
i8
58
68
55
18
63
68
50
18
56
68
67
18
7»
68
66
18
74
68
60
19
61
69
61
»9
62
69
70
•
19
60
69
70
»9
7*
69
68
»9
64
69
66 -
20
61
70
71
20
64
70
64
20
58
70
•62
20
58
70
5»
20
64
70
61
21
69
71
60
21
63
71
62
21
67
71
. 61
21
57
71
62
21
5!
71
59
22
62
7*
66
22
63
7*
63
22
67
7*
68
22
65
71
64
22
66
7a
64
23
65
73
61
23
64
73
55
»3
55
73
57
23
67
73
60
a3
62
73
69
»4
65
74
58
24
81
74
57
24
64
74
69
24
53
74
62
24
61
74
65
»5
63
75
64
15
66
75
70
*5
54
75
61
»5
61
75
66
a5
56
75
63
26
70
76
56
26
66
76
55
26
61
76
63
26
64
76
54
26
59
76
57
*7
54
77
58
27
71
77
65
17
79
77
62
27
58
77
76
a7
61
77
64
28
67
78
65
28
54
78
67
28
61
78
69
28
74
78
66
28
66
78
69
*9
55
79
64
»9
59
79
67
29
57
79
57
29
61
79
61
29
58
79
62
30
il
80
60
30
59
80
65
30
63
80
69
30
56
80
63
30
54
80
73
3»
81
59
31
67
81
69
31
63
81
60
31
58
81
57
31
67
81
60
3».
56
82
55
3a
53
82
68
3»
71
82
67
3*
66
82
67
32
71
82
60
33
59
83
65
33
66
83
72
33
69
83
66
33
61
83
63
33
55
83
69
34
65
84
61
34
59
84
57
34
58
84
77
34
58
84
68
34
55
84
62
35
61
85
63
35
63
85
59
35
59
85
70
35
64
85
7a
35
68
85
75
36
62
86
67
36
62
86
65
36
68
86
53
36
65
86
54
36
65
86
58
37
60
87
59
37
57
87
58
37
66
87
5a
37
61
87
66
37
66
87
60
38
55
88
71
38
57
88
61
38
49
88
66
38
67
88
55
38
59
88
58
39
64
89
59
39
59
89
74
39
72
89
54
39
62
89
71
•
39
•
59
89
64
40
72
90
59
40
62
90
59
40
67
90
65
40
63
90
61
40
61
90
H
41
63
9"
57
41
68
91
54
41
67
91
68
41
73
9»
62
41
54
91
67
4*
66
92
57
4a
59
92
60
42
63
92
58
42
66
92
62
4a
57
92
66
43
57
93
64
43
60
93
63
43
69
93
61
43
61
93
64
43
60
93
62
44
74
94
57
44
68
94
61
4^
68
94
59
44
60
94
62
44
60
94
66
45
70
95
71
45
58
95
62
45
57
95
69
45
67
95
63
45
60
95
63
46
63
96
59
46
53
96
70
46
61
96
64
46
74
96
58
46
55
96
61
47
54
97
53
47
63
97
55
47
51
97
57
47
62
97
64
47
74
97
60
48
61
98
64
48
61
98
59
48
66
98
63
48
70
98
62
48
62
98
68
49
59
99
66
49
70
99
75
49
62
99
58
49
62
99
62
49
59
99
51
Total
3121
3129
«
3x27
3»74
3161
3122
3171
3114
3092
3153
§n.]
BNTJMEEA.TION OF PEIMES IN THE FIRST NINE MILLIONS.
27
TABLE I. (continued).
Number of Pbimes in bach succbssivb group of 1000 Numbers, from to 9,000,000.
Ninth Million.
8,600,000
8,600,00C
)
8,700,000
8,800,000
8,900,000
to
to
to
to
to
8,600,000
8,700,001]
)
8,800,00C
*
8,900,000
9,000,000
o
68
50
64
63
50
66
62
50
70
59
50
54
57
50
65
I
60
51
73
I
66
5"
67
t
70
51
63
I
61
51
68
I
54
51
69
a
66
5»
72
2
58
5»
67
2
58
5a
57
2
63
Sa
62
2
64
sa
67
3
63
53
70
3
68
53
65
3
61
53
65
3
67
53
66
3
56
53
61
4
58
54
7»
4
64
54
69
4
60
54
5a
4
7a
54
70
4
57
54
63
5
. 70
55
56
5
59
55
64
5
69
55
64
5
49
55
56
5
56
55
62
6
57
56
51
6
58
56
75
6
67
56
69
6
58
56
53
6
69
56
60
7
63
57
67
7
71
57
57
7
62
57
62
7
65
57
60
7
63
57
55
8
68
58
64
8
64
58
61
8
55
58
7a
8
56
58
55
8
67
58
55
9
68
59
54
9
7«
59
60
9
59
59
63
9
70
59
63
9
64
59
58
lO
56
60
63
10
64
60
69
10
66
60
70
10
63
60
65
10
70
60
<3
ZI
71
61
59
IZ
69
61
69
II
69
61
64
II
5*
61
68
II
60
61
63
Z2
65
62
66
12
60
62
67
12
61
62
63
12
59
62
60
12
66
62
48
»3
67
63
61
13
60
63
53
13
67
63
69
»3
67
^3
59
13
66
^3
67
«4
58
64
57
14
68
64
56
14
65
64
61
H
7a
64
58
H
62
64
72
IS
60
55
54
15
5«
65
60
15
65
65
60
15
58
^5
59
»5
64
!?
60
i6
61
66
64
16
7»
66
63
16
66
66
64
16
62
66
59
16
64
66
66
17
65
67
60
»7
53
67
60
17
61
67
64
17
64
67
59
17
62
f7
60
i8
70
68
68
18
65
68
67
18
66
68
70
18
55
68
60
18
62
68
58
19
59
69
69
19
60
69
65
19
63
69
68
»9
58
69
68
»9
71
69
56
20
61
70
65
20
60
70
61
20
68
70
54
20
63
70
61
20
56
70
66
21
63
71
70
21
67
71
65
21
V
71
62
21
61
71
61
21
58
71
65
22
65
72
59
22
61
72
11
22
65
72
64
22
62
72
71
22
74
72
57
*3
71
73
74
23
63
73
»3
62
73
67
13
6z
73
70
23
63
'73
61
44
70
74
66
14
59
74
53
H
60
74
62
24
67
74
66
a4
64
74
58
as
f5
75
69
»5
64
75
55
25
69
75
62
as
71
75
53
*5
70
75
64
26
62
76
76
26
59
76
67
26
66
76
63
26
75
76
62
26
67
76
71
27
73
77
63
27
71
77
5a
27
62
77
53
17
63
77
70
27
59
77
58
28
55
78
66
28
63
78
70
28
56
78
53
28
63
78
67
28
64
78
1*
29
56
79
56
29
55
79
48
29
64
79
60
29
59
79
60
29
64
79
62
30
55
80
70
30
56
80
66
30
63
80
66
30
55
80
62
30
62
80
64
31
69
81
65
31
58
81
68
31
7*
81
74
31
67
81
56
31
54
81
66
3»
64
82
61
32
64
82
58
32
63
82
65
3*
60
82
66
32
60
82
68
33
63
83
58
33
64
83
70
33
62
83
69
33
57
83
57
33
70
83
i^
34
74
84
64
34
66
84
67
34
59
84
7»
34
59
84
49
34
61
84
61
35
58
85
54
35
64
85
71
35
53
85
64
35
59
H
68
35
It
85
68
36
61
86
70
36
65
86
59
36
59
86
58
36
61
86
75
36
86
55
37
58
87
55
37
62
87
67
37
67
87
71
37
70
87
65
37
66
87
69
38
66
88
61
38
68
88
60
38
62
88
60
38
51
88
70
38
6j
88
64
39
57
89
66
39
70
89
65
39
64
89
60
39
67
89
47
39
63
89
63
40
77
90
67
40
62
90
60
40
64
90
54
40
66
90
58
40
!«
90
55
4«
58
91
61
41
61
9'
65
41
65
91
55
41
57
91
60
41
60
91
tl '
4»
59
92
59
4»
54
92
68
42
58
92
64
4a
57
92
63
4a
58
9a
66
43
71
93
64
43
. 64
93
61
43
69
93
64
43
55
93
72
43
64
93
69
44
58
94
58
44
55
94
58
44
59
94
62
44
67
94
67
44
54
94
58
4S
61
95
63
45
58
95
62
45
55
95
58
45
63
95
63
45
67
95
69
46
68
96
70
46
1?
96
70
46
60
96
70
46
H
96
59
46
64
96
67
47
56
97
66
47
97
62
47
61
97
57
47
76
97
h
47
h
97
60
4«
67
98
53
48
• 56
98
61
48
68
98
5+
48
61
98
66
48
66
98
60
49
46
99
63
49
70
99
62
49
55
99
63
49
62
99
62
49
51
99
67
Total
3160
3166
3129
3152
3139
3160
3108
31Z2
3135
3»35
b2
26
INTEODTTCnON.
[§II.
TABLE I. (continued).
NUMBEK OF PeIMES IN BACH SUCCESSIVE GROUP OP 1000 NUMBERS, PROM TO 9,000,000,
Ninth Million.
8,000,000
8,100,00c
1
8,200,00c
►
8,300,00C
f
8,400,000
to
to
to
•
to
to
8,100,000
8,200,000
1
8,300,00(1
1
8,400,000
8,600,000
o
76
50
60
57
50
70
54
50
67
74
50
56
69
50
62
I
53
51
57
I
67
51
72
I-
70
51
64
1
63
5«
l^
I
71
51
65
2
59
5*
67
2
53
5*
57
2
61
5*
58
2
62
5*
65
2
59
5»
64
3
64
53
75
3
67
53
65
3
61
53
62
3
68
53
57
3
54
53
68
4
66
54
58
4
55
54
66
4
58
•54
58
4
60
54
65
4
55
54
61
5
61
55
65
5
62
55
t^
5
71
55
64
5
60
55
54
5
62
55
«5
6
62
56
80
6
64
56
61
6
75
56
56
6
64
56
65
6
57
56
63
7
66
57
66
7
67
57
71
7
62
57
62
7
56
57
61
7
68
57
63
8
66
58
61
8
58
58
60
8
62
58
70
8
65
58
67
8
69
58
73
9
64
59
63
9
69
59
61
9
65
59
57
9
59
59
64
9
53
59
57
xo
58
60
61
10
' 63
60
65
10
66
60
59
10
69
60
5»
10
63
60
60
II
59
61
65
ji
63
61
62
II
68
61
56
II
61
61
58
II
66
61
66
12
67
62
72
12
71
62
71
12
60
62
7a
12
73
62
57
12
68
62
64
13
57
63
64
13
56
63
64
13
64
f3
57
13
64
63
59
»3
64
63
1^
H
60
64
58
14
54
64
65
H
67
64
57
14
60
64
62
«4
61
64
67
15
64
65
67
15
65
65
63
"5
59
65
61
15
60
65
65
15
71
65
61
i6
62
66
54
16
70
66
59
16
66
66
62
16
61
66
59
16
56
66
58
17
63
67
71
17
66
67
66
17
64
67
58
»7
54
67
62
17
56
67
54
i8
58
68
55
18
63
68
50
18
56
68
67
18
7»
68
66
18
74
68
60
19
61
69
61
19
62
69
70
19
60
69
70
X9
7»
69
68
19
64
69
66
20
61
70
7»
20
64
70
64
20
58
70
•62
20
58
70
5»
20
64
70
61
21
69
71
60
21
63
71
62
21
67
71
. 61
21
57
71
62
21
5!
7"
59
22
62
7*
66
22
53
7»
63
22
67
72
68
22
65
7a
64
22
66
7»
64
»3
!5
73
6x
»3
64
73
55
»3
55
?3
57
23
67
73
60
23
62
73
69
H
65
74
58
H
81
74
57
24
64
74
69
14
53
74
62
»4
61
74
65
*5
63
75
64
»5
66
75
70
*5
54
75
61
»5
61
75
66
15
56
75
63
26
70
76
^t
26
66
76
55
26
61
76
63
26
64
76
54
26
59
76
57
27
54
77
58
27
71
77
65
27
79
77
62
27
58
77
76
»7
61
77
64
28
67
78
65
28
54
78
67
28
61
78
69
28
74
78
66
28
66
78
69
19
55
79
64
29
59
79
67
29
57
79
57
29
61
79
61
29
58
79
62
30
59
80
60
30
59
80
65
30
63
80
69
30
56
80
63
30
54
80
73
31
64
81
59
31
67
81
69
31
63
81
60
31
58
81
57
31
67
81
60
3».
56
82
55
3»
53
82
68
32
71
82
67
32
66
82
67
3*
71
82
60
33
59
83
65
33
66
83
7»
33
69
83
66
33
61
83
63
33
55
83
69
34
65
84
61
34
59
84
57
34
58
84
77
34
58
84
68
34
55
84
62
35
61
85
63
35
63
85
59
35
59
85
70
35
64
85
7»
35
68
85
75
36
62
86
67
36
62
86
65
36
68
86
53
36
65
86
54
36
65
86
58
37
60
87
59
37
57
87
58
37
66
87
5*
37
61
87
66
37
66
87
60
38
55
88
71
38
57
88
61
38
49
88
66
38
67
88
55
38
59
88
58
39
64
89
59
39
59
89
74
39
7»
89
54
39
62
89
71
•
39
59
89
64
40
7»
90
59
40
62
90
59
40
67
90
65
40
63
90
61
40
61
90
64
4«
63
91
57
41
68
9'
54
4>
67
9'
68
41
73
91
62
41
54
91
67
4»
66
92
57
42
59
92
60
4a
63
92
58
41
66
92
62
43t
57
92
66
43
57
93
64
43
60
93
63
43
69
93
61
43
61
93
64
43
60
93
62
44
74
94
57
44
68
94
61
44
68
94
59
44
60
94
62
44
60
94
66
45
70
95
71
45
58
95
62
45
57
95
69
45
67
95
63
45
60
95
63
46
63
96
59
46
|5
96
70
46
61
96
64
46
74
96
58
46
55
96
61
47
54
97
53
47
63
97
55
47
51
97
57
47
62
97
64
47
74
97
60
48
61
98
64
48
61
98
59
48
66
98
63
48
70
98
62
48
62
98
68
49
59
99
66
49
70
99
75
49
62
99
58
49
62
99
62
49
59
99
51
Total
3x21
3129
•
3127
#
3*74
3161
3122
3171
3114
3092
3153
§n.]
ENTJMEEATION OF PEIMES IN THE FIEST NINE MILLIONS.
27
TABLE I. (continued).
Number of Primes in bach successiye geoup of 1000 Numbers, from to 9,000,000.
Ninth Million.
8,500,000
8,600,OOC
1
8,700,000
8,800,000
8,900,000
to
to
to
to
to
8,600,000
8,700,001]
)
8,800,000
8,900,000
9,000,000
o
68
50
64
63
50
66
62
50
70
59
50
54
57
50
^5
I
60
51
73
I
66
5"
67
I
70
51
63
I
61
51
68
I
54
5»
69
2
66
5*
72
2
58
5a
67
2
58
5*
57
2
63
5»
62
2
64
5*
67
3
63
53
70
3
68
53
65
3
61
53
65
3
67
53
66
3
56
53
61
4
58
54
72
4
64
54
69
4
60
54
5»
4
7»
54
70
4
57
54
63
5
. 70
55
56
5
59
55
64
5
69
55
64
5
49
55
56
5
56
55
62
6
57
56
51
6
58
56
75
6
67
56
69
6
1*
56
53
6
69
56
60
7
63
57
67
7
71
57
57
7
62
57
62
7
65
57
60
7
63
57
55
8
68
58
64
8
64
58
61
8
55
58
7a
8
56
58
55
8
67
58
55
9
68
59
54
9
71
59
60
9
59
59
63
9
70
59
63
9
64
59
58
lO
56
60
63
10
64
60
69
10
66
60
70
10
63
60
65
10
70
60
«3
II
71
61
59
II
69
61
69
II
69
61
64
II
5»
61
68
II
60
61
^2
12
65
62
66
12
60
62
67
12
61
62
63
12
59
62
60
12
66
62
48
13
67
63
61
13
60
63
53
13
67
63
69
13
67
63
59
13
66
63
67
14
58
64
57
H
68
64
56
14
65
64
61
14
7*
64
58
»4
62
64
72
IS
60
55
54
15
56
65
60
15
^5
65
60
15
58
65
59
15
64
65
60
i6
61
66
64
16
7«
66
63
16
66
66
64
16
62
66
59
16
64
66
66
17
65
67
60
17
53
67
60
17
61
67
64
17
64
67
59
17
62
67
60
i8
70
68
68
18
65
68
67
18
66
68
70
18
55
68
60
18
62
68
58
19
59
69
69
»9
60
69
65
»9
63
69
68
«9
58
69
68
19
71
69
56
20
6t
70
65
20
60
70
61
20
68
70
54
20
63
70
61
20
56
70
66
21
63
71
70
21
67
71
^J
21
57
71
62
21
61
71
61
21
58
7"
65
22
65
72
59
22
61
72
65
22
!5
72
64
22
62
72
71
22
74
72
57
13
71
73
74
a3
63
73
56
*3
62
73
67
23
61
73
70
43
63
'73
61
a4
70
74
66
24
59
74
53
a4
60
74
62
a4
67
74
66
H
64
74
58
»5
^J
75
-59
»5
64
75
55
25
69
75
62
^5
71
75
53
a5
70
75
64
26
62
76
76
26
59
76
67
26
66
76
63
26
75
76
62
26
67
76
71
27
73
77
63
27
71
77
5*
27
62
77
53
17
63
77
70
27
59
77
58
28
55
78
66
28
63
78
70
28
i^
78
53
28
63
78
67
28
64
78
58
29
56
79
56
29
55
79
48
29
64
79
60
29
59
79
60
29
64
79
62
30
55
80
70
30
56
80
66
30
63
80
66
30
55
80
62
30
62
80
ii
3«
69
81
65
31
58
81
68
31
7*
81
74
31
67
81
56
31
54
81
66
3»
64
82
61
32
64
82
58
3»
63
82
65
3»
60
82
66
3a
60
82
68
33
63
83
58
33
64
85
70
33
62
83
69
33
57
83
57
33
70
83
t^
34
74
84
64
34
66
84
67
34
59
84
7»
34
59
84
49
34
61
84
61
35
58
85
54
35
64
85
71
35
53
85
64
35
59
85
68
35
tt
85
68
36
61
86
70
36
65
86
59
36
59
86
58
36
61
86
75
36
86
^5
37
58
87
55
37
62
87
67
37
67
87
71
37
70
87
65
37
66
ll
69
38
66
88
61
38
68
88
60
38
62
88
60
38
51
88
70
38
63
88
64
39
57
89
66
39
70
89
65
39
64
89
60
39
67
89
47
39
61
89
63
40
77
90
67
40
62
90
60
40
64
90
54
40
66
90
58
40
6S
90
55
41
58
91
61
41
61
9»
65
41
65
91
55
41
57
91
60
41
60
9"
tl '
42
59
92
|9
4»
54
92
68
4a
1^
92
64
4»
57
92
63
4*
58
9>
66
43
71
93
64
43
. 64
93
61
43
69
93
64
43
55
93
72
43
64
93
69
44
58
94
S8
44
55
94
58
44
59
94
62
44
67
94
67
44
54
94
58
45
61
95
63
45
58
95
62
45
55
95
58
45
63
9|
63
45
67
95
69
46
68
96
70
46
i?
96
70
46
60
96
70
46
6j
96
59
6^
46
64
96
67
47
56
97
66
47
97
62
47
61
97
64
47
76
97
47
65
97
60
48
67
98
53
48
• 56
98
61
48
68
98
48
61
98
48
66
98
60
49
46
99
63
49
70
99
62
49
55
99
63
49
62
99
62
49
5«
99
67
Total
3160
•
3166
3129
3152
3139
3160
3108
3112
3135
3135
e2
28
INTEODFCnON.
[§u.
TABLE 11.
NuMBEB OF Primes m each successive gboup of 10,000 Numbebs, from to 9,000,000.
1,000,00c
)
2,000,000
3,000,000
4,000,000
to
to
to
to
to
1,000,000
2,000,00c
)
3,000,000
4,000,000
6,000,000
X230
50
753
753.
50
73»
705
50
677
670
50
651
660
50
\n
X
1033
51
770
X
7x9
51
702
X
691
51
675
X
659
51
663
X
658
51
638
a
983
5*
764
2
73*
5*
69 X
2
693
5*
697
2
663
5»
630
2
668
52
646
3
958
53
747
3
70X
53
686
3
690
53
672
3
657
53
663
3
677
53
631
4
930
•
54
750
4
731
54
698
4
671
54
670
4
671
54
654
4
681
54
648
5
924
55
750
5
698
55
714
5
696
55
672
5
657
55
668
5
643
55
659
6
878
56
747
6
716
56
681
6
694
56
678
6
664
56
670
6
653
56
673
7
902
57
769
7
722
57
701
7
674
57
699
7
525
57
640
7
670
57
650
8
876
58
763
8
706
58
693
8
686
58
693
8
686
58
667
8
653
58
640
9
879
59
747
9
738
59
676
9
674
59
676
9
654
59
669
9
665
59
(>s$
xo
861
60
763
xo
736
60
719
10
6l?
60
653
10
676
60
692
10
663
60
66s
IX
848
61
751
XX
716
6x
694
XI
61
681
II
677
61
670
XI
628
61
656
iz
858
62
729
X2
7x8
62
71X
12
697
62
672
12
666
62
686
12
652
62
645
»3
851
63
733
»3
7«9
63
692
13
673
63
68 X
13
666
63
652
13
632
63
651
H
838
64
757
«4
700
64
692
14
693
64
689*
»4
658
64
638
14
661
64
651
15
835
65
733
X5
728
65
701
15
712
65
695
15
677
65
650
15
662
^1
616
x6
814
66
745
x6
734
66
7x6
16
667
66
662
16
668
66
662
x6
671
66
665
17
828
67
754
17
726
67
702
17
690
67
665
'Z
663
67
702
17
6si
67
666
i8
68
75»
18
735
68
675
18
679
68
681
x8
691
68
638
x8
673
68
667
19
8x4
69
728
«9
713
69
713
'9
664
69
686
«9
675
69
681
«9
647
69
644
20
823
70
763
20
676
70
696
20
701
70
679
20
686
70
iss
20
670
70
65a
2X
811
71
723
2X
741
71
685
21
660
7«
695
21
674
71
671
21
644
71
653
22
819
7»
760
22
693
7*
689
22
695
7»
645 •
22
672
72
687
22
663
72
<43
23
784
73
74a
23
693
73
*3
680
73
657
*3
687
73
658
23
628
73
663
»4
823
74
707
»4
7H
74
706
»4
683
74
672
*4
649
74
649
24
664
74
644
as
793
75
740
*5
713
75
684
»^
688
75
671
*5
662
75
667
25
660
75
64»
26
805
76
755
26
7x8
76
679
26
701
76
684
26
677
76
632
26
644
76
6SS
a7
790
77
735
27
709
77
700
*7
694
.77
666 '
*7
666
77
J55
27
656
77
6a8
28
792
78
738
28
722
78
689
28
662
78
663
28
670
78
659
28
632
78
657
»9
773
79
745
29
689
79
713
29
685
79
684
29
648
79
657
29
649
79
638
30
803
80
73*
30
709
80
704
30
690
80
690
30
673
80
656
682
30
662
80
«S«
31
808
8x
733
31
703
8x
•
672
3«
662
81
695
31
660
81
31
666
81
65s
3»
796
82
745
3»
7»3
82
7x8
3*
672
82
667
3*
677
82
674
32
641
82
631
33
778
83
729
33
705
83
675
33
671
83
704
33
645
83
646
33
656
83
678
34
795
84
7»7
34
692
84
701
34
667
84
671
34
675
84
677
34
635
84
634
35
780
85
725
35
7«3
85
707
35
690
85
654
35
623
85
650
35
640
85
645
36
765
86
753
36
709
86
703
36
691
86
672
36
688
86
646
36
653
86
669
37
778
87
728
37
7*3
87
689
37
662
87
651
678
37
667
87
65 X
37
661
87
636
38
767
88
732
38
695
88
697
38
705
88
38
669
88
661
38
662
88
669
39
793
89
719
39
741
89
69 X
39
681
89
662
39
662
89
681
39
«35
89
679
40
754
90
752
40
679
90
689
40
660
90
68 X
40
677
90
651
40
641
90
651
4»
776
91
708
4«
68x
91
697
4«
692
91
664
4«
666
9'
^*
41
679
91
634
4*
77»
9»
740
4»
717
9»
7"
4*
672
92
671
4*
672
92
■^
42
683
92
653
43
779
93
713
43
723
93
684
43
657
93
680
43
685
93
43
656
93
655
44
765
94
720
41
702
94
692
44
70X
94
649
H
670
94
678
41
672
94
650
45
75*
95
7x1
45
701
95
684
45
687
95
653
45
646
95
643
45
655
95
640
46
765
96
73*
46
7x6
96
673
46
668
96
694
46
654
96
638
46
660
96
683
47
782
97
717
47
705
97
670
47
672
97
658
47
637
97
668
47
646
683
97
661
48
761
98
7x0
48
706
98
689
48
687
98
671
48
636
98
634
48
98
654
49
772
99
7"
49
698
99
714
49
674
99
687
49
668
99
642
49
638
99
641
Total
41539
36960
35<>53
34780
34H»
33744
33334
32995
32802
«
32567
§u.]
ENXTMERATION OF FBDIES IN THE FIBST NINE MILLIONS.
29
TABLE II. (continued).
«
Number of Pbimes in each successive group of 10,000 Numbers, from to 9,000,000.
5,000,000
•
6,000,000
7,000,000
8,000,000
to
to
•
to
to
6,000,000
7,000,000
8,000,000
f
9,000,000
o
639
50
638
66x
50
642
629
50
626
637
50
641
I
$3?
51
623
I
638
51
640
I
630
51
626
I
609
5»
632
a
658
5*
633
2
664
5*
646
2
654
5*
613
2
63 X
5*
64X
3
t^l
53
667
3
601
53
629
3
648.
53
638
3
60s
53
625
4
628
54
6i9
4
627
54
641
4
657
54
653
4
639
54
621
5
668
55
634
5
666
55
640
5
628
55
645
5
652
55
^43
6
639
56
655
6
6,5
56
634
6
594
56
6J
6
628
56
62X
7
648
57
636
7
653
57
639
7
660
57
7
623
57
654
8
655
58
641
8
628
58
659
8
640
58
637
8
619
58
624
9
646
59
657
9
644
59
632
9
629
59
644
9
607
59
624
lb
632
60
633
10
652
60
641
10
663
60
657
10
6x9
60
642
XI
641
61
638
II
641
61
662
II
609
61
617
XX
633
61
•
626
12
65s
62
638
12
636
62
615
12
611
62
649
12
651
62
622
13
660
<J3
643
13
637
63
620
13
617
63
611
13
602
63
637
H
639
64
624
14
637
64
640
14
654
64
642
H
622
64
602
15
644
U
641
»5
645
65
613
15
640
65
632
15
648
65
651
i6
645
635
16
658
66
648
16
629
66
626
x6
535
66
629
17
631
67
638
17
627
67
630
17
615
67
630
17
625
67
59*
i8
641
68
635
18
619
68
618
18
630
68
640
18
648
68
65X
19
648
69
679
«9
650
69
651
»9
638
69
643
19
6x8
69
629
20
659
70
635
20
644
70
655
20
632
70
612
20
fi29
70
623
ai
649
71
667
21
602
71
634
21
630
71
645
21
630
71
649
22
650
7*
632
22
653
72
646
22
636
72
630
22
623
72
629
»3
647
73
632
^3
632
73
634
23
625
73
607
»3
638
73
624
H
673
74
654
H
637
74
641
»4
653
74
622
24
631
74
614
»S
630
75
636
15
650
75
661
li
615
li
626
a5
618
75
637
26
649
76
617
26
647
76
619
647
612
26
619
76
'653
47
653
77
637
27
636
77
632
27
642
77
647
17
629
77
600
28
629
78
637
28
664
78
620
28
636
78
611
28
634
78
659
29
654
79
640
29
660
79
633
»9
632
79
633
29
622
79
6xx
30
654
80
654
30
646
80
632
30
628
80
638
30
631
80
620
31
637
81
632
31
634
81
653
31
630
81
633
3»
646
81
610
3*
643
82
627
3*
649
82
634
3*
637
82
624
32
618
82
645
33
649
648
83
641
33
633
83
631
33
616
83
642
33
618
83
606
34
84
6s7
34
630
84
633
34
621
84
647
34
658
84
627
35
642
85
6»7
35
637
85
663
35
642
85
632
35
627
85
607
36
642
86
63s
36
626
86
660
36
643
86
634
36
608
86
««s
37
653
87
641
37
609
87
623
37
^n
87
647
37
623
87
641
38
654
88
677
38
628
88
«5«
38
628
88
641
38
636
88
615
39
640
89
64s
39
645
89
631
39
619
89
626
39
620
89
634
40
630
90
648
40
630
90
637
40
620
90
630
40
617
90
607
41
640
91
667
41
643
91
641
41
641
91
616
41
643
9«
647
4a
657
92
649
4»
635
92
615
4*
633
92
634
4»
611
9»
«39
43
650
93
646
43
611
93
652
43
636
93
629
43
619 ,
93
618
44
63s
94
^33
44
633
94
642
44
656
94
640
44
602
i
94
614
45
659
95
639
45
659
95
^3i
45
615
95
604
45
641
95
61S
46
641
96
633
46
632
96
648
46
610
96
634
46
6i»
96
613
47
623
97
636
47
628
97
632
47
647
97
638
47
633
97
620
4«
651
98
644
48
637
98
630
48
632
98
634
48
639 '
98
653
49
6$%
99
625
•
49
639
99
«37
49
6x1
99
61S
49
6a8
1
99
634
Total
32287
32049
31908
J1891
31623
3»535
31364
4
31396
30
INTEODFCnON.
[§ll.
The following four tables give the number of primes in each group of a hundred thousand,
a quarter of a million, half a million, and a million from to 9,000,000, with differences. It
will be seen that there are thirty-four instances in which the number of primes in a group
100,t)00 exceeds the number in the preceding group, that there are four instances in which the
number in a group of 250,000 exceeds the number in the preceding grpup, and that even when
the group is as large as 500,000 the numbers of primes do not continually diminish, there being
one instance of an increase :
NUMBEE OF FbIMES IN EACH SUCCESSIVE GEOUP OF 100,000, FROM TO 9,000,000.
LiinitB.
Number
of Primes.
Difference.
LimitB.
Number
of Primes. •
Difierenoe.
o- 100,000
9593
4,500,000-4,600,000
6493
- - .
120
1 00,000- 2 00,000
8392
-1201
4,600,000-4,7 00,000
^5523
+
30
200,000- 300,000
8013
—
379
4,700,000-4,800,000
6475
—
48
300,000- 400,000
7863
—
160
4,800,000-4,900,000
<5554
-f
79
400,000- 500,000
7678
—
185
4,900,000-5,000,000
6522
—
32
500,000- 600,000
7560
—
118
5 ,000,000-5 » ' OOjOOO
6458
—
64
600,000- 700,000
7445
—
115
5,100,000-5,200,000
6436
—
22
700,000- 800,000
7408
—
37
5,200,000-5,300,000
<5493
+
57
800,000- 900,000
7323
—
85
5,300,000-5,400,000
6462
—
31
900,000- 1 ,000,000
7224
—
99
5,400,000-5,500,000
6438
—
24
1,000,000-1,100,000
7216
—
8
5,500,000-5,600,000
6402
—
36
1,100,000-1,200,000
7225
+
9
5,600,000-5,700,000
6404
+
2
1,200,000-1,300,000
7081
—
144
5,700,000-5,800,000
6387
—
17
1 ,300,000-1 ,400,000
7103
+
22
5 ,800,000-5 ,900,000
6436
+
49
1 ,400,000- 1 ,500,000
7028
—
75
5,900,000-6,000,000
6420
—
16
1 ,500,000-1 ,600,000
6973
—
55
6,000,000-6,100,000
<5397
—
23
1,600,000-1,700,000*
7015
+
42
6,100,000-6,200,000
6402
+
5
1 ,700,000-1 ,800,000
6932
—
83
6,200,000-6,300,000
6425
+
23
1 ,800,000-1 ,900,000
6957
+
25
6,300,000-6,400,000
6337
—
88
1 ,900,000-2 ,000,000
6903
—
54
6,400,000-6,5 00,000
6347
• +
10
2 ,000,000-2 , 1 00,000
6874
—
29
6,500,000-6,600,000
6402
+
55
2,100,000-2,200,000
6857
—
17
6,600,000-6, 7 00,000
^33^
—
64
2,200,000-2,300,000
6849
—
8
6,7 00,000-6,800,000
637s
+
37
2,300,000-2,400,000
6791
—
58
6,800,000-6,900,000
641 1
+
36
2,400,000-2,500,000
6770
—
21
6,900,600-7,000,000
6365
•^
46
2,500,000-2,600,000
6809
+
39
7 ,000,000-7 , 1 00,000
6369
+
4
2,600,000-2,700,000
6765
—
44
7,100,000-7,200,000
6306
—
63
2,700,000-2,800,000
6716 .
—
49
' 7,200,000-7,300,000
6348
+
42
2,800,000-2,900,000
6746
+
30
7 ,300,000-7 ,400,000
6299
—
49
2,900,000-3,000,000
6708
—
38
7,400,000-7,500,000
6301
+
2
3,000,000-3, 1 00,000
6676
—
32
7,500,000-7,600,000
6305
+
4
3,100,000-3,200,000
6717
+
41
7,600,000-7,700,000
6347
+
42
3,200,000-3,300,000
6691
^
26
7,700,000-7,800,000
6245
—
102
3,300,000-3,400,000
6639
—
52
7 ,800,000-7 ,900,000
6364
+
119
3,400,000-3,500,000
661 1
—
28
7 ,900,000-8,000,000
6274
—
90
3,500,000-3,600,000
657s
—
36
8,000,000-8, 1 00,000
6250
—
24
3,600,000-3,700,000'
6671
-h
96
8, 1 00,000-8,200.000
6301
+
51
3,700,000-3,800,000
• 6590
—
81
8,200,000-8,300,000
6283
—
18
3,800,000-3,900,000
6624
+
34
8,300,000-8,400,000
6285
+
2
3 ,900,000-4,000,000
6535
—
89
8,400,000-8,500,000
6245
—
40
4,000, 000-4, 1 00,000
6628
+
93
• 8,500,000-8,600,000
6326
+
81
4, 1 00,000-4, 20O9OOO
6540
—
88
8,600,000-8,700,000
6281
—
45
4,200,000-4,300,000
6510
—
30
8,700,000-8,800,000
6299
+
.18
4,300,000-4,400,000
6511
-h
1
8,800,000-8,900,000
6220
—
79
4,400,000-4,500,000
6613
. +
102
8,900,000-9,000,000
6270
■f
50
§u.]
BNUMEpATION OF PHTMEfl HT THE FIEST NINE MILLIONS.
31
NUMBBB OF FBDIBS IN EACH SUCCESSIVE GBOUP OF 250,000, FBOM TO 9,000,000.
•
Limits.
Number
of Primee.
Bifferenoe.
Limits.
•
Number
of Pnmes.
o- 250,000
22,045
4,500,000-4,750,000
16,271
- 94
• 250,000- 500,000
19,494
-2551
4,7 50,000-5,000,000
16,296
+ 25
500,000- 750,000
18,700
- 794
5,000,000-5,250,000
16,172
-124
7 50,000-1 ,000,000
18,260
- 440
5,250,000-5,500,000
16,115
- 57
1 ,000,000-1 ,2 50,000
i7»97i
- 289
5,500,000-5,750,000
16,026
- 89
1 ,2 50,000-1 ,500,000
17,682
- 289
5,750,000-6,000,000
16,023
- 3
1,500,000-1,750,000
17,455
- 227
6,000,000-6,2 50,000
15,967
- 56
1,750,000-2,000,000
17,325
• - 130
6,2 50,000-6,500,000
15,941
- 26
2,000,000-2,250,000
17,150
- 175
6,500,000-6,7 50,000
15,950
+ 9
2,250,000-2,500,000
16,991
- 159
6,750,000-7,000,000
15,941
- 9
2,500,000-2,7 50,000
16,922
- 69
7,000,000-7,250,000
15,851
- 90
2 , 7 5 0,000-3 ,000,000
16,822
- 100
7,250,000-7,500,000
J5,772
- 79
3,000,000-3,2 50,000
16,761
- 61
7,500,000-7,750,000
15,768
- 4
3,250,000-3,500,000
16,573
- 188
7,750,000-8,000,000
15,767
- 1
3,500,000-3,750,000
16,566
- 7
8,000,000-8,2 50,000
15,712
- 55
3>7 50,000-4,000,000
16,429
- 137
8,250,000-8,500,000
15*652
- 60
4,000,000-4, 2 5 0,000
16,437
+ 8.
8,500,000-8,7 50,000
15,746
+ 94
4,250,000-4,500,000
16,365
- 72
3,750,000-9,000,000
15,650
- 96
NUMBEB OF PbIMES IN EACH SUCCESSIVE GEOUP OP 500,000, PEOM TO 9,000,000.
Limits.
•
Number
of Primes.
•
Difference.
Limits.
Number
of Primes.
Difierenoe.
9
o- 500,000
500,000-1 ,000,000
1 ,000,600-1 ,500,000
1 ,500,000-2,000,000
2,000,000-2 , 5 00,000
2 , 5 00,000-3 ,000,000
3,000,000-3,500,000
3,500,000-4,000,000
4,000,000-4,500,000
41539
36960
35653
34780
34141
33744
33334
32995
32802
-4679
-1307
- 873
- 639
- 397
- 410
- 339
- 193
•
4,500,000-5,000,000
5,000,000-5,500,000
5 1500,000-6,000,000
6,000,000-6,500,000
6,5 00,000-7 ,000,000
7 ,000,000-7 , 5 00,000
7,500,000-8,000,000
8,000,000-8,500,000
8,500,000-9,000,000
32567
32287
32049
31908
31891
31623
31535
31364
31396
- 236
- 280
- 238
- 141 .
- 17
- 268
- 88
- 171
+ 32
NuMBEB OF Primes fob the complete Millions.
1 1 I i,^^„.^,_,^,.,,^_^^.„^^
First million
Second „
Third „
Fonrtli „
Fifth „
Sixth „ ,
Seventh „
Eighth „
Ninth „ ,
Number of
•
Difference.
prmies.
•
•
78,499
70,433
-8066
67,885
-2548
66,329
-1556
65,369
- 960
64,336
-1033
63,799
- 637
63,158
- 641
62,760
- 398
.^
32
INTEODTJCTION.
[§ll.
Thp number of primes in the first three millions is 216,817, in the second three millions is
196,034, and in the third three millions is 189,717 ; the successive diminutions being 20,783
and 6317.
The total number of primes in the first million, the first two millions, the first three
millions, &c., are as follows :
Limits.
Kamber of PrimeB.
0-1,000,000
78,499
0-2,000,000
148,932
0-3,000,000
216,817
0-4,000,000
283,146
0-5,000,000
348,51s
0-6,000,000
412,851
0-7,000,000
476,650
0-8,000,000
S39i8o8
0-9,000,000
602,568
In Burckhardt's tables there is no mark of any kind corresponding to the number 2,882,6 99,
this being, no doubt, caused by the type having slipped back. This number was found to be
divisible by 19. Owing to a similar accident there is nothing to show whether the number
6,036,637 in Base's tables is a prime. This number was found to be a prime.
In all the present enumerations the numbers 1 and 2 have both been regarded as primes.
Gauss gave 168 as the number of primes in the first thousand numbers, and no doubt he excluded
unity. If unity be excluded, the numbers of primes in the first century, thousand, myriad,
hundred thousand, and million are 25, 168, 1229, 9592, and 78,498 respectively.
Envmeration according to Numbers of Primes in Centuries,
The following ninety tables (pp. 33-55) show the number of centuries in each group of
10,000, or myriad, from to 9,000,000, each of which contains no primes, the number of
centuries each of which contains one prime, two primes, &c. Each table refers to 100,000
numbers, and the Sioman numerals I., II., ... X. at the head of the columns denote the first,
second, • . . tenth myriad of the group. Thus the column headed I. of the first table shows that of
the hundred centuries which form the group to 10,000 there are two which contain each seven
primes, there are three which contain each eight primes, and so on, there being one which
contains twenty-six primes. Similarly the column headed II. shows that in the second myriad
there is one century which contains five primes, one which contains six primes, five which contain
each seven primes, and so on. The last column gives the total for the 100,000 numbers. Thus
of the thousand centuries between and 100,000 three contain each three primes, six contain
each four primes, &c., and so on.
The number at the foot of each column is the total number of primes in the group of
numbers to which the colunm has reference : thus between and 10,000 there are 1230 primes,
between 10,000 and 20,000 there are 1033, and so on. The numbers at the foot of the columns
in these tables are the same as those which are given in Table II. (pp. 28 and 29), and in the
table on page 30.
The tables for the fourth and fifth millions have already been published in the Introduction
to these Millions.
§11.]
BNUMEEATION OF PRIMES IN THE FIRST NINE MILLIONS.
33
•
•
'■
to 100,000.
•
•
«
•
100,000 to* 200,000
•
•
•
tfft
Number of oenturies each of which contains n primes.
Number of centuries each of which contains n primes.
• •
!••
I.
n.
111.
IV.
V.
VI.
vn.
VIM.
IX.
X.
Total.
*
n.
I.
n.
III.
IV.
V.
VL
Vll VllL
TX.
X.
Total.
o
.
c
1
.0
•
•
•
1
I
o»
1
2
b
2
•
S
o
.0
6
I
I
I
3
3
p
I
I
I
2
2
7
4
o
2
I
I
2
6
4
I
*
2
I
I
3
4
I
15
5
o
I
X
I
2
2
I
3
3
4
18
5
6
5
9
I
4
I
8
4
2
7
47
6
I
I
5
5
4
6
6
6
5
39
•
6
10
14
9
10
12
15
14
8
14
10
116
7
2
5
8
6
J4
9
16
16
13
17
106
.
7
14
II
14
16
12
16
19
13
14
16
145
8
3
10
18
•
10
»5
20
H
»5
22
149
8
13
20
18
»7
18
22
H
20
.la
20
184
9
7
16
»9
23
— ^»
23
*4
21
18
17
185
9
22
22
13
16
28
22
»7
19
20
17
196
10
12
z8
•
17
H
21
'5
14
15
16
.177
10
16
II
19
14
12
8
9
20
22
15
146
11
19
18
>5
10
II
7
12
8
9
124
11
10
3-
7
7
7
II
12
6*
4
."
78
12
•
»9
^3
12
13
9
3
8
5
8
104
12
. 7
6
10
3
4
2
.2
.4
4
42
13
9
. 5
6
5
2
5
3
4
2
46
13
I
4
I
3
I
3
2
2
I
18
14
7
2
2
•0
2
I
2
I
17
14
•
I
I
I
3
15
12
I
I
14
15
I
I
2
16
6
I
I
0*
8
•
17
2
•
•
•
2
•
21
4
1
•
.
26
I
b
«
c
1
•
Number 1
[of primee [
1230
1033
983
95.8
930
924
878
902
876
879
9593
Number 1
of primes j
861
848
858
851
^38
835
814
845
828
814
^39^
200,000 to 300,000.
300,000 to 400,000.
s»
Number of centuries each of whibh contains n primes.
•
n.
Number of centuries each of which contains n primes.
•
1»«
I.
n.
in.
IV.
V.
VI.
VU.
vni.
TX.
X.
Total
I.
n.
III
IV.
V.
•
VI.
VII
.VIII
. IX.
X.
Totol.
•
•
•
•
*
1
•
.
I
•
1
1
•
2
•
2
I
•
I
2
8
2
2
I
I
2
I
3
I
13
.
3
2
I
2
I
2
3
z
0.
lH
4
I
2
3
2
4
6
I
3
2
24
4
2
2
I
4
6
6
7
6
3
3
40
•
5
' 2
4
.4
5
8
5
5
8-
8
6
55
5
5
5
6
IZ
8
8
8
5
6
10
72
6
•
16
10
12
14
10
14
18
II
II
18
134
6
12
»5
13
13
10
«5
12
9
17
.11
127
7
14
20
22
23
»5
17
8
»3
14
17
173
7
«9
13
28
»3
16
• •
20
14
»9
.16
H
172 .
8
20
22
«5
20
21
19
17
22
21
*5
202
8
16.
22
20
18
H
13
20
16
.*7
H
200
9
23
17
17
14
16
16
.20
14
16
14
167
9
20
21
II
»7
16
15
18
18 •
15
20
171
10
9
13
17
17
»9
12
9
10
15
10
131
. 10
12
12
II
II
10
9
II
15
10
10
111
11
7
5
6
2
7
5
zi
6
5
5
69
11
5
5
4
8
3
4
7
5
3
5
49
12
4
3
2
I
3
6
2
4
2
2
29
12
5
I
4
2
3
7
I
3
2
I
29
13 ,
1
I
z
2
2.
7-
43
3
I
I
3
2
z
11
14
I
I
2
I
5
14
I
I
•
•0
2
•
'
1
15
I
I
2
Number 1
of primes*/
823
811
819
•
784
823
793
805
790
79»
773
8013
Number "1
of primes/
803
808
796
778
795
780
765
778
767
793
7863
84
INTEODTrcnON.
[§11.
400,000 to 600,000.
600,000 to 600,000.
am
Number ot oeotories eaeh of whidi ooataini n primei.
«
•
Number of oenturies each of which oonteiiiB n primes. 1
If*
1.
n.
m.
IV.
V.
VL viL vm.
IX
X
TotaL
•
%m
1.
n.
HI.
IV.
V.
VL Vll.
VIM.
IX.
X
TotaL
o
o
p
•
1
o
I
1
•
1
•
•
2
o
o
I
I
I
I
4
2
r
I
I
3
3
I
I
3
I
I
2
2
2
13
3
«
2
•
3
4
2
5
•
I
2
20
4
8
4
4
3
3
5
3
I
5
3
39
4
I
I
3-
9
4
5
7
5
3
6
44
6
II
7
12
7
6
12
7
7
II
II
91
5
12
9
8
3
10
10
II
9
7
8
•
87
6
•
II
i8
ID
21
17
15
»7
16
16
12
153
6
«3
19
14
20
12
16
It
»3
12
12
142
7 .
«7
15
"5
7
»7
«9
-21
»9
»5.
20
176
7
26
16
26
»5
16
20
8
'5
28
a3
193
8
i8
i8
a3
28
12
14
*5
»9
12
20
189
8
»5
17
16
18
«9
17
22
22
20
19
185
9
»7
'4
12
13
H
15
10
17
16
16
144
«
9
H
14
18
14
»7
12
14
18
17
15
153
• 10
9
II
8
II
8
8
6
10
9
6
86
10
12
17
IZ
8
to
8
»3
II
8
It
110
11
5
5
8
5
7
6
5
6
9
7
63
11
3
4
I
8
4
7
4
2
5
2
40
12
2
5
3
\
4
3
3
2
•
4
3
33
12-
I
I
2
2
2
3
4
I
16
13
I
I
2
'2
I
7.
13
I
I
I
I
I
I
•
I
7
14
O
o
•
I
I
2
Number "1
of primes J
754
776
772
779
765
75a
765
782.
761
772
•
7678
Number 1
of primes/
753
770
764
747
750
7S0
•
747
769
763
f
747
7560
600,000 to 700,000.
700,000 to 800,000.
n.
*
Number of oeaturies each of which contains
n primes.
I.
n.
*
111.
•
iv.
V.
VI.
VII.
vni
IX
X
Total.
p
•
1
2
I
I
I
2
I
I
7
. 3
I
3
I
I
I
3
•
I
5
16
4
2
6
3
4
6
10
4
3
5
43
5
II
10
14
10
8
8
7
12
14
II
105
6
18
»1
10
18
14
17
16
>4
»3
12
145
7
16
H
18
*5
«9
22
16
a4
.*5
»9
208'
• 8
18
12
23
14
16
8
28
21
17
16
173
9
14
16
»9
17
II
»9
12
14
12
21
155
10 .
>3
6
4
6
16
9
5
2
9
8
78
11
6
f
5
4
4
4
6
5
6
I
47
12
I
5
I
I
I
4
2
1
16
13
2
• I
I
I
I
6
14
I
1
•
•
•
•
•
•
Number \
of primes/
763
751
729
733
757
733
745
754
75»
728
7445
n.
1
Number of centuries each of which contains n primes. 1
I.
n.
in.
IV.
V.
VI.
VTT.
VUL TX
X
TotaL
.0
1
2
I
I
I
I
I
5
3
4
2
I
2
2
i
3
I
1
2
19
4
.6
7
4
3
9
7
3
5
4
5
53
5
5
»5
9
12
9
7
8
13
8
II
97
6
14
9
20
16
20
16
17
•
14
22
14
162
7
16
21
17
21
18
27
14
23
20
16
193
8
«9
19
H
19
18
II
*3
16
20
*5
194
9
16
10
2
10
12
14
17
«4
12
12
119
10
10
9
12
6
4
6
10
9
5
8
79
ii
10
5
7
6
5
8
3
2
•
3
5
64
12
2
2
3
I
I
2
3
2
I
17
13
2
I
I
I
I
6
14
.0
•
.
16
I
1
16
.
17
6
I
1
Number 1
of primes/
763.
723
760
74*
707
740
755
735
738
745
7408
I
§11.]
ENUMERATION OF PSIMES IN THE FIBST NINE MILLIONS.
85
800,000 to 900,000.
•
Mm
Number of centuries each of which oontaiiiB
» primes.
>l»
I.
n.
lU.
IV.
V.
VL
VII.
vin.
TX.
X.
Total.
o
1
1
o
o-
2
I
"0
2
•0
4
3
I
4
4
4
I
I
I
I
2
21
4
4
4
.4
6
9
5
7
4
3
63
6
II
8
16
13
9
14
9
Il-
15
114
6
14
as
20
17
10
17
14
2*0
ls
15
165
7
25
«7
10
II
a4
17
16
18
21
23
182
8
13
J7
»4
18
«5
21
15
9
*4
20
176
9
18
II
16
II
'4.
14
18
17
«5
II
145
10
7
7
12
8
6
8
12
5
«
6
76
11
4
5
5
2
6
7
3
«
4
4
44
• 12
2
I
2
2
1
2
I
13
13
1
2
I
5
14
I
I
•
2
Number 1
of primes j
73a
733
745
729
7*7
7^S
753
728
732
719
73*3
•
900,000 to 1,000,000.
•
Number of centuries each of which contains n primes.
•
n*
•
I.
n.
in.
IV.
V.
vrvii.
viir.
IX.
X
Total.
1
•
'
2
I
I
I
I
4
3
3
I
2
I
2
I
4
2
16
4
I
7
5
9
4
7
6
•
I
8
7
55
6
17
12
II
8
•
H
9
10
14
10
10
115
6
9
19
17
14
16
2»
18
19
•
21
26
179
7
»3
16
21
27
»3
22
21
»4
22
19
218
8
*5
19
16
16
17
16
«7
«5
13
15
169
9
II
13
14
H
10
16
II
7
»3
10
119
10
9
7
5
7
II
4
6
7
3
5
64
11
2
2
3
2
I
3
6
5
6
4
34
12
2
a
4
I
I
2
2
I
2
17
13
I
•0
2
I
I
I
I
2
9
14
•
15
•
I
.0
1
Number "1
of primes j
75»
708
740
713
7»o.
711
732
717
710
721
7224
1,000,000 to 1,100,000.
•
•
«
•
Number of centuries each of which contains n primes.
I.
n.
III.
IV.
V.
VI.
VII.
vin.
IX.
X.
Total.
•0
1
I
1
2
I
I
1
I
4
3
4
2
2
3
I
•
2
3
I
21
4
2
8
S
4
3
6
8
4
8
53
5
II
10
8
18
12
10
II
12
15
8
115
6
13
14
i8*
21
16
22
"9
15
17
15
170
7
27'
17
13
22
»4
44
16
^3
20
21
217
a
'9
»9
21
8
H
15
21
16
15
16
164
9
II
13
9
»3
14
14
•
II
12
II
17
125
10
8
6
8
5
9
6
10
9
73
11
6
6
4
6
3
3
I
5
39
12
*
I
1
2
I
I-
2
2
12
13
1
I
.
I
I
1
6
Number
of primes j
753
719
73*
701
731
698
716
722
706
738
<
7216
1,100,000 to 1,200,000.
Number of centuries each of which contains n primes.
•
A.
«
1
I.
n.
in.
•
IV.
V.
VI.
•
Vll.
Vin.
DL
X.
Total.
•
1
I
1
2
I
I
I
> 2
5
3
•
4
3
I
3
3
3
3
2
I
28
4
5
6
1
9
5
9
3
4
5
6
6
58
6
8
II
10
II
10
12
9
II
12
8
102
6
H
20
17
20>
»7
17
16
17
»7
«5
170
7
21
21
22
20
21
21
21
17
29
»7
220
8
22
II
10
II
»9
20
16.
19
9
»9
156
9
«3
16
16
17
7
II
18
15
12
II
136
10
9
6
10
9
8
8
6
8
3
9
76
11
I
4
2
4
2
2
4
5
• 9
2
35
12
I
I
I
2
2
I
2
10
13
I
I
z
I
I
I
6
14
I
a
I
•
2
Number 1
of primes
736
716
7.1
«
7«9
700
7.8
734
726'
735
713
7225
f2
36
INTEODXJOTION.
[§IL
1,200,000 to 1,300,000.
1,300,000 to 1,400,000.
Number of oenturies eaob of which containB f» primee.
•
». .
1
1
*
•
I.
II.
III.
IV.
V.
VL VII. vm.
TX.
X
Total.
o
*
•
1
o
I
I
2
2
2
%
I
I
. 6
3
3
2
4
5
4
3
I
4
3
3
32
4
T
7
7
3
5
7
12
2
3
10
63
6
15
12
12
»S
10
H
9
IS
6
12
120
6
i6
14
13
'?
17
16
16
IS"
20
•
14
160
•
7
•
»4
15
*S
H
21
20
IS
22
24
14
214
8
17
19
•16
II
17
IS
22
18
19
14
168
9
8
12
7
10
12
*
n
14
13
13
9
111
10
3
II
10
S
10
4
.s
3
9
10
73
11»
3
6
3
2
3
s
3
6
I
3
35
12
I
I
I
I
3
I
I
9
13
I
I
I
2
5
14
I
0.
1
15
q
•
16
o
I
1
Number "1
of primee
676
744
69J
693
724
713
718
709
7»»
689
*
7081
•
Number of oeaturies «aoh dL yrhioh Qontnios n primes.
n*
1
1
1
I.
n.
lU.
IV.
V.
VI. VIL VIII.
LSL
X.
Total.
•
•
1
•
I
1
2
I
I
X
X
I
2
•
I
. I
9
3
3
I
I
3
2
2
2.
S
19
4
3
10
7
II
6
8
7
6
6
s
69
5
17
y
II
II
IS
12
8
8
14
xo
119
6
IS
13
17
14
20
18
23
17
16
19
172
7
22
19
26
18
21
16
16
28
19
22
207
8
14
22
14
16
14
13
21
17
IS
16
162
9
17
II
12
14
12
9
12
7
ij
II
118
10
5
6
2
II
•s
IS
S
8
6
9
72
11
4
2
7
I
I
3
3
6
3
3
33
12
. I
2
I
2
2
2
I
I
3
15
13
I
I
.0
•
1
3'
14 •
.
15
•
i
I
1
Number 1
of primes j
709
703
713
705
692
713
709
723
69s
741
7103
1,400,000 to 1,600,000.
1,600,000 to 1,600,000.
M
Number of centuries each of which contains n primes.
•
I9»
•
•
•
I.
n.
IIJ.
IV.
V.
VL
Vii.
VIII.
IX.
X.
Total.
a
0.
•
0.
1
X
I
X
X
X
5
2
I
X
2
2
I
7
3
3
3
2
I
2
2
•
4
2
19
4
8
8
8
4
6
7
9
9
•
5
8
72
5
-^1
9
7
14
14
II
14
IS
16
13
ISO
6
21
22
20
18
19
19
II
16
16
19
181
7
17
H
18
18
13
»4
18
XI
IS
22
180
8
12
16
28
17
H
14
17
18
»3
14
183
9
12
XI
15
6
XO
13
12
7
8
98
10
7
•
2
9
7
8
9
8
9
73
11
2
2
jf
2
3
6
5
4
4
34
12
I
2
'2
2
X
X
2
.1
16
13
X
2
Number 1
of primes j
•
679
681
717
7*3
702
701
7.?
705
706
698
7028
n.
Number of centuries each of which contains
n primes.
I.
n.
UT.
IV.
V.
VI.
Vll.
VIII.
iX.
X.
TotaL
i
.0
1
•
X
I
2
•
2
•
X
I
2*
2
I
3
10
3 •
2
4
2
2
3
s
3
6
X
28
4
8
5
5
7
9
13
6
\o
7
7
77
5
8
»?
9
13
IX
9
12
15
II
17
124
6
16
20
as
*
21
20
X2"
*s
14
»3
22
198
7
19
21
18
19
i.S
IS
13
19
II
19
172
8
19
12
IS
18
•
IS
17
xo
17
IS
I*.
•
150
9
16
14
16
12
8
17
IS
6
•
II
124
10
8
3
5
4
9
7
5
10
10
63
11
S
2
2
3
3
3
2
4
29
12
4
X
3
X
X
4
X
17
13
I
2
a
•0
6
Number T
of primes j
731
702
691
686
698
714.
681
701
693
676
6973
§11.]
EWMTTRATIOM OF PBTMES IN THE ilBST NINE MILLIONS.
1,600,000 to 1,700,000. 1,700,000 to 1,800,000.
37
AB
•
Number of oenturiefl moh of whioh oontaiiis
• •
•
n primes.
'*"
•
1
1
L
XL
HI.
IV.
V.
VL VII. vin.
•
TX.
X
Total.
o
I
•0
1
1
o
.0
2
2
2
I
3
I
•I
I
2
2
11
3
3
3
3
4
4
2
2
4
4
29
4
7
4
9
7
7
10
4
4
10
5
67
6
10
II
8
12
II
II
13
12
18
IS
121
6
I8
22
15
19
»5
II
14
19^
10
16
159
7
SA
«S
14
•
21
*5
18
22
^4
^4
18
203
8
H
»3
as
"5
17
16
17
IS
«3
19
174"
9
8
12
18
12
12
20
12
8
14
13
129
10
1
7
• 2
4
8
4
•7
II
7
4
9
63
11
7
3
2
t
3
I
3
2
2
3
27
12
2
I
I
I
4
' 9
13
I
1
•
I
1
3
14
I
•
1
1
•
15
o
I
1
Number 1
of primes j
719
694
7"
692
692
701
716
702
675
713
1
7015
1,800,000 to 1,900,000.
Number of centuries eiioh of which contains n primes.
••
I.
U.
•
III.
IV.
V.
VI.
VU.
VITT.
TX.
X,
Total.
o»
•
r
1
•
2
I
I
2
.0
2
10
3
3
2
I
I
I
I
2
3
22
4
6
5
10
7
6
5
5
. 8
71
5
14
15
10
•
10
12
19
17
12
135
6
13
20
H
22
19
16
«9
13
176
7
»S
26
18
•
21
•
21,
20
22
19
204 '
8
15
>9
13
22
«5
19
IS
II
160
9
10
7
19
9
8
10
12
10
115
10
9
4
8
6
13
4
6
8
10
74
11
2
4
3
5
2
S
2
23
12
2
I
I
1
2
2
I
•
1
•
b
10
«
Number 1
of primes/
704
672
718
«7S
701
707
763
689
697
691
hS7
1
2
3
4
5
6
'7
8
9
10
11
12
13
14
15
Numoer \
of primes J
Number of centuries eacdi of which contains n primes.
I.
n.
in.
IV.
V.
3
7
13
17
*3
10
19
3
I
o
o
4
9
IS
16
21
16
II
I
3
o
o
O I
o
6
19
22
22
II
8
8
3
o*
3
6
16
22
15
IS
II
7
o
o
VI.
vn.
VUL
IX
Total.
o o
3
S
12
20
22
16
15
2
o
I
4
8
IS
14
19
IS
10
o
o
3
•6
21
IS
19
13
12
6
o
o
S
6
13
13
•
21
18
14
9
I
o
o
3
10
13
18
17
19
10
S
4
I
o
o
o
2
7
IS
17
IS
13
IS
II
2
2
o
o.
o
30
70
152
174*
194
146
125
61
26
10
•
5
1
695
685
691
689
706
684
679
700
689
713
$932
1 ,900,000 to- 2,000,000.
Number of centuries each of whibh contains n primes.
ft.
I.
II.
in.
IV.
V.
VI.
fn.
Vill.
IX.
X.
Total.
1
I
1
2
I
I
2
I
5
3
4
3
I
2
10
I
. 3
4
4
2
34
4
•5
4
4
6
4
9
7
10
II
7
67
5
12
18
IS
18
.11
12
II
16
ro
12
135
6
19
20
18
16
17
44
40
20
19
10
183
7
21
20
*3
»7
20
16
as
17
21
31
221
8
16
10
16
14
14
19
17
IS
8
20
149
9
IS
14
8
7
9
II
8
8
IS
6
101
10
5
6
8
7
II
4
S
S
6
6
63
11
2.
4
6
2
3
• 2
I
3
2
S
30
12
I
I
m
2
I
5
13
I
I
I
3
14
I
I
I
«
3
Number 1
of primes/
689
697
711
684
692
684
673
670
689
714
6903
38
INTEODUCTION.
[§n.
2,000,000 to 2,100,000.
Number of oenturies each of whioh oontaina
fipiim^
M.
,
1
»
*
I.
n.
III.
IV.
V.
VL
viL vin.
IX
X.
Total.
o
•
•
.
1
o
•
2
I
3
2
9
2
2
.1
2
2
I
10
3
3
•3
5
2
•
2
•
3
4
5
2
3
32
4
7
8
9
4
8
5
6
7
9
6
09
5
13
16
9
15
»3
10
12
«3
9
15
•
119
• 6
»5
20
«3
16
26
»3
»5
14
17
18
197
7
lO
22
13
»S
»3
*3
18
a3
»5
22
204
8
13
«7
15
22
12
."3
12
16
18
>9
157
9
i6
'5
II
4
II
12
II
H
^i
10
116
•10
xo
2
8.
6
II
7
7
3
3
6
63
11
2
I
•
5
3
o-
I
3
3
3
•0
21
12
o
2
I
2
2
X
8
13
•
I
I
2
Number 1
of prixneB j
705
691
693
690
671
696
V
694
674
686
674
6874
2,100,000 to 2,200,000.
Number of oenturiee each of which oo&tauiB n primea.
«.
*
1
.
I.
n.
TIT.
IV.
V.
VI.
VIL VIII.
TX.
X
TotaL
1
X
X
2
2
I
I
X
'
X
X
I
X
2
9
3
5
3
3
I
4
2
3
•
2
4
27
4
7
7
2
»
«3
5
3
9
7
9
8
70
6
12
»4^
20
x6
16
il
17
12
12
14
145
6
12
Tfi
17
14
x6
*5
16
»3
2X
»9
183
7
19
»4
18
*3
26
22
x8
22
22
17
201
8
22
21
•
20
20
12
16
'9
xo
X2
15
167
9
12
10
8
7
9
10
8
>3
x6
17
110
10
6
5
6
2'
7
7
8
5
2
4
52
11.
3
2
3
I
4
2
x
2
18
12
X
X
• X
X
X
2
2
9
13
X
X
I
•
X
4
14
X
2
3
r
Number 1
of primes/
699 683
697
673
6^3
7X2
667
690
679 664
6857
2,200,000 to 2,300,000.
2,300,000 to 2,400,000.
n.
1
2
3
4
5
6
7
8
9
10
11
12
Number of centuries each of which contains n primes.
I.
n.
Number 1
of primes j
5
8
12
17
19
12
14
7
5
X
o
2
9
18
17
12
9
6
2
X
m.
IV.
V.
VI.
VIL
VIIL
IX
I
4
5
x6
12
as
«9
6
6
4
2
I
7
5
13
18
21
9
6
4
X
o
2
10
II
18
X«
18
»9
X
2
o
2
I
7
10
20
20
II
3
2
o
o
2
7
16
23
13
16
7
2
o
o
2
7
17
as
17
9
S
A
2
3
2
6
IS
21
19
16
13
o
2
2
TotaL
2
.a
9
9
22
18
22
.7
3
•S
X
701
660
695
680
683
688
701
694
662
685
2
9
29
73
138
179
205
168
113
44
30
10
6849
n.
1
2
3
4
5
6
7
8
9
10
11
12
13
Number of centuries each of which contains n primes.
L
n.
III.
IV.
V.
VL
vn.
VIIL
IX.
o
X
Number 1
of primes J
4
8
13
•
22
13
13-
o
X
I
12
18
16
as
9
9
o
X
690
662
2
XO
13
21
20
x6
6
o
o
3
9
14
20
20
21
7
o
X
4
8
16
«7
13
17
14
S
o
o
3
7
12
21
X6
IS
14
s
o
o
2
3
XO
26
26
IS
8
8
o
o
7
13
13
II
18
IS
13
6
3
o
o
o
4
7
10
14
19
21
8
2
o
o
3
9
16
16
IS
IS
IS
s
2
TotaL
672
671
667
690
691
662
70s
•
4
10
33
86
135
177
194
157
113
54
29
681 . 6791
§ II.]
ENUMEEATION OP PMMES Df THE MEST NINE MILLIONS.
89
2,400,000 to 2,5b0,000.
2,500,000 to 2,600,000.
•
Number of oonturieB each of which oontaiiis n primes.
*
». •
1
I.
n.
ni.
IV.
V.
VL Vll.
VIII.
TX.
X
TotaL
o
o
1
o
o
•
I
1
•
2
%
o
I
I
2
3
9'
3
4
6
4
4
I
5
3
5
3
2
37
4
IX
8
7
1
7
9
7
10
8
•
6
4
78
6
13
H
17-
19
15
12
17
II
II
17
146
6
i8
i6
21
20
il
22
18
18
20
22
193
7
21
i6
19
20
t?
17
22
22
?9
17
190
8
lO
x6
H
«4
21
17
xo
17
16
x6
151
9.
8
«3
9
IX
7
12
6
II
12
13
102
10
•
9
4
5
4
6
4
8
6
.7
3
56
11
%
6
3.
I
■4
1
I
4
I
2
I
25
12
I
o
I
I
X
I
2
7
13
o
I
•
1
2
I
5
Number ">
of prunes J
66o
692
672
657
701
687
668
672
687
674
6970
Number of centuries each of which contains n primes.
lim
.
I..
JLL
III.
IV.
V.
VI.
Vll.
vm.
TX.
X.
Total.
1
X
I
•
<;
I
3
2
X
2
I
I
5
3
4
6
6
2
2
I
2
8
•
4
35
4
7
18
9
7
8
6
10
7
8
8
88
5
xo
XX
7
15
«S
*3
16
16
xo
•
»3
136
6
22
14
20
20
20
»5
18
19
20
*5
193
7
M
17
12
20
22
21
13
»4
«3
12
178
8
x8
15
20
10
x6
19
x6
21 '
19
19
173
9
8
10
13
7
•
9
7
6
12
6
10
88
10
2
5
7
10
5
4
7
4
9
4
57
11
«
x
5
4
2
2
I
2
3
•
5
25
12
X
I
I
2
2
x
'2
I
2
13
13
2
X
I
2
6
Nnmber \
of primes j
677
675
697
672
670
672
678
699
*
693
676
6809
2,600,000 to 2;700,000.
2,700,000 to 2,800,000.
Number of centuries each of which contains n primes.
«
fl.
•
I.
n.
in.
IV.
V.
VI.
VII.
vm.
IX.
X.
Total.
I
o*
*
1
1
2
2
4
2
x
2
I
2
I
2
I
10
3
3
6
2
2
4
3
3
3
2
28
4
9
6
6
12
3
7
6
8
6
7
70
5
II
15
14
II
13
17
»3
16
18
21
159
6 •
26
»7
•
14
18
*3
»9
20
*4
«9
»5
195
7
.*3
II
27
20
21
21
21
14
22
21
201
8
14
*3
16
13
15
x6
III
8
12
»3
141
9
9
10
13
IS
10
xo
6
xo
5
8
96
10
3
6
5
5
2
8
2
5
8
xo
54
11
I
I
x
3
X
4
4
5
2
22
12
3
2
4
I
I
4
I
X
17
13
I
1
14
I
1
•
Number 1
of primes
653
681
672
681
689
69s
662
665
681
686
6765
•
•
Number of centuries each of which contains n primes.
1.
n.
HL
IV.
V.
VI.
VU.
VIII.
TX.
•
3t.
ft
Total.
. 1
I
p
I
2
2
2
2
2
6
I
• 7
3
4
5
€
5
4
4
4
3
3
5
43
4
9
7
15
7
8
9
8
8
II
II
93
5
10
H
14
H
13
12
13
17
12
18
137
6
»4
18
15
28
19
20
15
21
18
17
195
7
18
22
»5
20.
a4
x6
*3
»9
22
9
188 ,
8
13
10
«3
12
15
13
20
19
15
15
145
9
9
9
xo
4
XI
t^
9
7
8
8
87
. 10
6
9
6
7
5
8
7
2
7
. 10
67
11
3
3
4
2
2
3
I
6
24
•
•12
I
3
X
I
I
I
I
9
13
X
I
2"
14
15
16
17
I
1
*
Number \
of primes j
679
695
645
657
672
671
•
684
666
663
684
6716
40 .
INTEGDTTCJTION.
[§ II.
2,800,000 to 2,900,000.
2,900,000 to 3,000,000.
Number of oenturiee each ofwhioh oontiuiii
n primes.
n»
I.
n.
a
in.
IV.
V.
VL
VII. VlJi. IX.
X.
Total
•
•
1
•
I
I
2
2
o
%'
4
5
2
I
I
15
S
%
4
3
4
3
I
3
2
4
SO
•
4
•
9
6
9
7
8
10
II
9
8
84
6
•
H
14
7
16
19,
16
«9
»5
W
6
i8
17
20
13
23
18
16
18
15
21
179
7
44
22
21
20
20
»7
22
*3
21
22
222
8
>3
18
9
20
'3
12
9
12
12
»4
132
9
10
17
12
12
8
6
14
9
10
II
109
•
10
7
6
3
5
7
. 5
5
8
3
53
11
3
3
5
I
2
I
2
18
12
2
I
2
I
7
13
O
I
2
2
I
6
14
I
I
2
Number 1
ofprimee
690
695
66f
704
671
654
672
6S$
678
662
6746
mm
•
, Number of eentories each of whioh oontains
fl primes.
•
I.
n.
111.
IV.
V.
VI.
Vll.
VIII
IX.
X
Total.
0.
1
I
2
2
I
X
2.
2
X
3
13
3
• ^
3
2
8
6
4
4
44
4
• 7
6
6
7
•9
• 6
4
64
5
20
12
14
18
12
15
19
»4
II
152
6
17
21
22
18
18
15
26
22
17
187
7
18
30
18
22
22
26
»3
15
a3
214
8
IC
II
12
12
17
11
II
14
»9
135
9
10
9
12
13
9
6
12
II
103
10
4
5
6
3
6
%
7
4
. 58
11
I
3
I
2
I
4
15
12
I
I
2
2
I
11
13
6
I
«
1
14
c
15
•
.0
I
1
•
Number 1
of primes J
681
664
671
680
649
653
694
658
671
687
6708
. 3,000,000 to 3,100,000.
3,100,000 to 3,200,000.
•
fl.
Number of oentunes each of which contams fi primes.
L
n.
m.
IV.
V.
VI.
vn.
vm.
IX.
X.
Total
•
1-
•
2
2
•
2
I
3
2
2
I
I
11
3
2
4
3
2
5
4
3
7
2
37
4
•9
7
12
9
5
9
7
4
8
81
5
16
II
12
22
ti
10
17
9
16
140
6
21
22
18
17
26
»5
20
21
19
206
7
17
*3
20
18
18
18.
18
22
.»5
183
8
•
19
«5
17
18
H
19
16
16
15
168
9
5
9
4
9
9
6
H
12
15
94
10
6
4
8
4
8
5
4
4
7
54
11
4
I
4
2
.1
3
2
19
12
X
I
2
4
13
*
X
.
1
Number 1
of primes J
670
659
663
657
671
657
664
695
686
654
6676
^ ' '
•
Number of oenturiee each of which oontaiiis n primes.
II.
L
II.
III.
IV.
V.
VI.
Vll.
Vlil.
EL
3t.
•
TotaL
•
I
1
1 •
2
•
X
3
2
I
2
I
4
2
I
13
8
2'
7
2
2
5
5
3
31
4
4
10
9
8
9
9
9
7
14
87
5
18
16
9
«7
12
20
12
16
12
«4.
146
6
20
H
»9
as
21
12
H
«3
•
19
16
173
7
17
13
20
18
19
*4
16
»9
18
24
198
•
8
IS
12
18
15
21
II
»3
17
20
17
169
9
13
13
7
7
14
8
9
13
7
96
10
6
5
9
4
•7
3
5
4
4
52
11
3
I
4
I
5
X
2
2
19
12
2
I
X
2
2
9
13
I
2
14
•
•
I
1
Number .1
of primes J
676
677
666
666
658
677
668
663
691
675 6717
§11.1
ENTTMEBATION OF PBTMEfl IN THE FIBST NIKE MILLIONS.
41
3^0,000 to 3^0,000.
3,300,000 to 3,400,000.
Number of oenturies each of wbloh oontalnB ft primas.
Number of oeaturiee each of which oontainB n priiDflB.
o.
n.
I.
n.
111.
IV.
V.
VI. vii vm.
TX.
X.
Total.
I.
n.
111.
IV.
V.
VL
vn.
VUL
TX.
X.
Total
o
1
t
1
1
X
X
X
I
' 4
2
a
%
3
X
3
ft
X
ft
16
2
ft
X
ft
X
X
X
r
4
13
3
5
%
6
ft
3
ft
4
ft
X
6
S3
3
6
4
3
Xft
X
4
4
7
4
5
50
4
7
ID
4
7
XX
Xft
6
14
5
15
91
4
3
14
8
4
8
14
7
7
10
xo
85
5
10
x6
20
Xft
13
8
14
13
ftft
XX
139
5
»3
13
XI
19
»5
•17
13
9
13
xo
133
6
»3
i6
«5
19
17
ftO
ftX
ftft
ftft
»4
199
6
a4
H
18
x6
ftO
21
»5
x8
20
x8
194
7
i8
17
18
20
30
ftft
ftO
17
x6
»S
193
7
x8
16
*S
14
17
12
9
22
x8
20
171
8
13
i8
17
17
Xft
x8
Xft
8
15
xo
140
8
IS
17
XX
ftO
20
*9
20
a3
xo
«4
169
9
XI
lO
7
9
xo
8
XX
«5
6
xo
97
9
xo
XI
15
9
9
8
12
8
XX
9
102
10
8
5
4
7
ft
5
6
3
7
3
50
10
5
6
6
3
6
X
8
3
9
4
51
11
3
3
5
3
X
ft
ft
3
3
6
31
11
4
3
X
3
2
X
2
X
2
4
23
12
X
I
%
ft
ft
X
9
12
I
I
X
2
5
13
o
X
1
Number 1
of primes J
686
674
67a
687
649
662
677
666
670
648
6691
Number "1
of primes J
673
660
677
64s
67s
623
688
667
669
662
6639
3,400,000 to 3,600,000.
3,600,000 to 3,600,000.
Number of centuries each of which contains
n primes.
M.
L
n.
ra.
IV.
V.
VL
VIL
VIII.
iX.
X.
Total
1
X
X
X
3
2
2
X
2
2 '
2
2
14
3
2
8
9
2
2
5
7
6
5
48
4
12
9
12
7
12
6
7
12
5
89
5
«5
13
12
13
15
20
«4
14
9
IS
140
6
"3
II
13
16
19
16
21
17
26
20
172
7
*7
22
12
19
22
*3
H
26
18
*3
216
8
10
14
20
20
13
14
H
13
12
13
143
9
12
16
9
«4
12
4
8
8
10
7
100
10
6
4
9
2
4
3
3
2
6
43
11
3
X
4
3
3
3
I
3
2
24
12
I
X
2
6
13
X
X
2
Number 1
of primes/
677
666
672
685
670
646
654
637
636
668
661 1
Number of centuries each of which contains
f» primes.
n.
I.
IL
m.
IV.
V.
VI.
VII.
VUL
IX.
X.
Total
1
X
2
X
4
2
2
I
X
I
X
2
3
12
3
4
3
4
s
3
2
6
s
3
42
4
6
10
12
9
10
9
6
XI
7
89
5
23
14
17
XI
II
16
13
a3
12
x6
156
6
16
19
20
20
20
20
a4
22
15
12
188
7
»3
«9
17
21
18
17
22
12
18
»S
192
8
7
22
»7
18
17
16
«7
H
'4
149
9
10
7
6
17
10
6
8
II
18
9
102
10
6
I
3
J^
S
4
3
X
2
6
35
11
• 3
3
X
S
2
2
2
3
24
12
X
X
I
3
13
X
X
X
3
14
X
1
Number 1
of primes/
651
663
630
663
654
668
670
640
667
669
6S75
42
QTIBODXTOIION.
ttn.
M.
1
2
3
4
5
6
7
8
9
10
11
12
13
3,600,000 to 3,700,000.
Knmber of oenturiM eaob of whioh oontaixu n primes.
I.
n.
Komber 1
of primes/
>9
9
i6
5
m.
12
14
21
i6
i6
9
6
692
IV.
V.
8
»5
16
3*
12
12
4
23
x8
13
12
8
4
VI.
VIL
»9
18
22
670
686
652
9
5
638
8
13
*3
18
17
7
5
vin.
»9
18
12
22
9
6
t
2
6
x6
13
19
*4
9
6
3
o
IX
650
662
702
3
6
XI
14
21
18
xo
7
5
5
o
638
%
2
9
x8
15
13
2X
8
9
2
X
Total.
68 X
2
13
31
76
180
190
178
160
94
66
21
6
6
6671
3,700,000 to 3,800,000.
Number of centuries each of wfaieh contsans n primes.
Ik
I.
n.
m.
IV.
V.
VI.
VII.
vin.
EL
X.
TotaL
1
3
2
X
X
2
17
3
4
3
3
4
43
4
8
xo
5
7
77
6
14
13
9
x6
x6
IS
20
14
12
X7
146
6
x8
20
x8
19
22
*s
29
2X
17
20
209
7
23
23
19
20
X7
20
14
X5
X2
170
8
13
X4
H
x8
x6
17
X2
22
19
2X
176
9
14
8
9
7
2
X4
83
10
f
8
5
6
8
62
11
X
2
21
12
X
8
Number 1
of primes J
'"
671
687
658
649
667
632
655
659
657
6590
3,800,000 to 3,900,000.
3,900,000 to 4,000,000.
Number of centuries each of which contains
fi primes.
n*
L
n.
m.
IV.
V.
VI.
vn.
VIM.
IX
X
TotaL
1
X
X
2
4
2
3
3
X
3
2
12
8
3
X
7
X
4
6
6
8
3
4
43
4
X2
7
6
15
13
8
XI
7
xo
xo
99
6
x6
x8
9
«7
12
15
13
x8
X2
14
144
6
20
x8
*5
13
«9
22
19
x6
17
20
189
7
15
>5
x6
2X
13
17
17
x6
2X
18
169
8
x6
x6
«9
15
x8
15
12
XX
2X
14
167
9
xo
•9
13
9
xo
9
9
13
XX
9
102
10
3
5
2
2
9
6
7
5
X
5
46
11
3
3
3
2
X
X
3
4
2
3
26
12
X
3
2
2
8
13
X
I
X
3
Number "1
of primes J
656
682
674
646
677
650
646
6sx
661
681
6624
».
Number of centuries each of which contains
» primes.
1
I.
II.
III.
IV.
V.
VI.
vn. vni.
IX
X
TotaL
X
1
1
X
I
X
X
4
2
2
4
X
X
2
X
X
X
2
16
3
3
5
2
3
6
3
3
S
«
4
42
4
8
9
4
8
7
X2
XX
xo
9
IX
89
6
17
«3
16
17
XX
13
19
X2
20
18
166
6
28
x6
>9
2X
»9
30
2X
17
x8
20
209
7
XX
x6
»4
20
17
12
20
»3
x6
20
179
8
«4
«S
2X
«3
«5
«5
XX
17
xo
9
140
9
5
9
10
XX
8
4
8
9
XX
5
80
10
7
xo
2
4
9
6
5
5
6
6
60
11
2
X
X
4
X
5
14
12
X
2
2
X
X
7
13
X
X
2
14
X
X
2
Number 1
of primes j
651
658
675
648
678
643
638
668
634
642
«S3S
y — J
4,000,000 to 4,100,000.
1 w r *i
4,100,000 to 4,200,000.
^«#
w.
Hnmber of oenturies eaofa of which eontains n primes.
II.
Number of oentnxiee eaeh of which oontaioB n primes.
L
n.
HI.
IV.
V.
VL
vn.
VUL
TX.
X
TotaL
L
n.
TTT.
IV.
V.
VI.
VU.
vm.
a.
z.
1V>teL|
o
1
o
o
2
I
3
1
I
2
3
2
I
I
I
2
2
3
I
2
2
15
2
2
4
I
2
2 I
2
17
3
3
3
4
3
I
5
3
4
I
2
29
3
5
6
6
2
2
I
5
5
39
4
>3
II
6
8
10
«3
7
8
10
6
92
4
9
8
»5
6
7
10
14
5
90
5
lO
H
19
»7
II
H
13
12
12
20
142
5
13
'5
13
II
H
22 13
14
12
»9
156
6
92
%o
10
21
iS
16
21
20
28
18
201
6
16
»7
28
«3
21
«7
»9
»s
18
21
195
7
l6
*4
18
20
21
18
*S
43
as
»5
215
7
22
20
'7
22
»5
16
20
»7
27
*4
200
8
«9
12
«7
8
18
»5
14
12
7
II
133
8
'5
9
14
15
13
18
II
19
16
7
187
9
10
8
8
10
9
9
10
15
8
6
93
9
II
12
5
7
9
12
7
II
89
10
5
3
5
7
6
4
2
4
4
5
45
10
3
2
8
6
10
4
5
5
•
3
48
11
I
a
3
3
5
2
I
4
21
11
4
2
I
I
3
2
2
16
12
I
2
I
2
I
7
12
I
2
2
1
I
7
13
o
I
2
8
13
8
14
o
o
010
1
Number 1
lof primes J
66o
658
668
677
681
^43
653
670
653
665
6628
Number 1
of primes/
663
6a8
652
63a
661
662
671
651
673
647
6540
4,200,000 to 4,300,000.
4,300,000 to 4,400,000.
Number of centuries each of which contains n primes.
Number of centuries each of which contains n primes. 1
ff.
ft.
1
L
IL
III.
IV.
V.
VI.
vn.
VIM
IX.
X.
ToUl.
L
n.
III.
IV.
V.
VI.
vn.
VUl.
IX.
X.
Total
1
I
I
I
3
1
2
I
2
4
I
3
3
I
I
17
2
3
I
3
4
I
I
3
16
3
3
3
8
3
2
4
2
3
I
31
3
S
2
4
2
I
4
6
S
35
4
9
II
10
5
10
13
«3
9
14
16
110
4
7
8
II
13
9
ID
10
9
II
100
5
12
13
»4
»4
»7
»3
9
20
20
19
151
5
16
16
17
16
II
13
13
14'
1
14
<
143
6
»3
34
20
20
13
«5
»4
20
18
'9
206
6
20
21
22
20
29
20
»7^
M
H
212
7
'7
16
17
16
22
19
17
12
15
10
161
7
24
18
19
18
21
28
as
II
12
205
8
»9
18
20
20
18
»3
22
15
12
166
8
II
»9
14
13
«4
8
13
12
16
133
9
9
5
9
6
10
13
9
10
IS
93
9
10
7
7
II
7
7
II
16
10
94
10
5
9
I
6
4
I
3
2
3
37
-
10
4
7
7
4
3
6
4
4
2
44
11
I
2
2
2
2
I
2
2
17
11
3
1
1
I
2
2
2
2
15
12
I
I
2
I
5
12
I
I
2
13
I
2
13
I
1
14
I
1
Number 1
of primes J
670
644
663
628
664
660
644
656
632
649
6510
Number "1
of primes/
662
666
641
656
635
640
653
661
662
63s
6511
g2
44
INTEODTTOnON.
Kn,
4,400,000 to 4,600,000.
4,500,000 to 4,600,000.
It.
Number of centuries each of which containe n primee.
L
n. m. IV.
V,
VI.
VU.
vui.
IX.
X.
Total.
o
b
1
o
1
I
I
8
2
o
3
%
2
1
2
I
13
3
6
3
%
3
2
3
37
4
7
5
6
8
9
5
12
75
5
fti
18
»S
16
17
12
18
163
6
11
ft3
»3
»3
a7
21
z8
21
207
7
17
17
17
20
12
«9
28
17
190
8
1%
21
15
»3
17
20
16
'4
163
9
8
10
12
8
5
12
zo
96
10
5
4
5
3
5
3
2
37
11
»
z
I
2
2
I
2
18
12
o
I
2
I
6
13
o
I
2
14
o
I
1
Number 1
of primeej
641
679
683
656
672
655
660
646
683
638
6613
Number of centuries each of which oantains n primes. 1
II.
1
I.
n.
m.
IV.
V.
VI.
Vll.
viir
EL
X
TotaL
1
2
2
2
z
2
2
z
3
5
2
18
8
6
4
6
5
5
6
3
8
65
4
Z2
zz
9
7
zo
3
Z2
zo
96
6
zo
»5
z8
17
14
zz
Z2
Z2
6
132
6
19
2Z
17
2Z
17
22
2Z
17
»3
193
7
20
z8
17
17
19
»3
z8
20
22
192
8
14
«7
13
z8
'7
14
z8
Z2
20
Z2
165
9
»4
7
zo
zo
9
Z2
ZO
8
8
97
10
z
3
6
2
3
5
6
3
4
35
11
3
2
z
3
z
1
z
3
19
12
z
z
2
2
6
Number 1
of primes J
653
638
646
63Z
648 659
673
650
640
655
«493
4,600,000 to 4,700,000.
4,700,000 to 4,800,000.
11.
Number of centuries each of which contains n primes.
L
n.
111.
IV.
V.
VL
VU.
vm.
EX.
X.
Total.
z
z
2
1
1
z
z
3
2
2
1
2
2
2
I
2
2
14
3
7
5
8
z
5
6
4
5
3
4
48
4
10
7
8
6
zo
13
9
9
4
7
83
5
zo
2Z
Z2
z8
16
2Z
«5
zz
19
19
162
6
13
Z2
20
27
20
20
»5
»5
20
21
193
7
z8
H
ZO
21
19
Z2
2Z
H
z8
20
187
8
20
H
»3
ZO
Z2
IS
13
z8
«3
13
141
9
Z2
zz
Z2
ZO
9
6
9
8
Z2
8
97
10
3
1
3
4
5
4
8
6
5
3
42
11
4
3
z
z
z
1
2
2
2
3
20
12
I
z
z
I
z
z
6
13
z
z
2
Number 1
of primes/
662
656
645
65Z
6sz
616
665
666
667
644
6523
Number of centuries each of which contains n primes.
II.
I.
n.
III.
rv.
V.
VI.
Vll.
VIM.
IX
X
Total.
Q
1
z
Z
z
fi
3
2
2
2
Z
3
2
2
3
2
z
20
3
4
6
S
2
8
3
7
2
5
49
4
9
zz
zo
zz
zz
zz
9
14
Z2
106
5
14
zz
2Z
14
14
IS
Z2
13
ZO
z6
140
6
22
z6
Z2
21
H
Z2
22
26
z8
IS
188
7
z6
z8
2Z
17
z6
z6
»3
z6
»3
»S
191
8
19
IS
Z5
ZZ
14
z6
zz
8
17
Z2
138
9
9
H
8
Z2
Z2
7
12
7
9
97
10
3
4
6
3
4
6
6
5
4
47
11
z
z
3
2
2
z
z
12
12
z
z
2
z
z
z
z
9
13
z
Q
1
Number 1
of primes J
652
653
643
663
644
642
655
628
657
638
fi47S
§11.]
ENUMEBATION OF FBIMES IN THS FIBST NINE MILLIONS.
46
.4,800,000 to 4,900,000.
4,900,000 to 5,000,000.
Number of oentories each of which oontftixu n primes.
w»
L
n.
III.
IV.
V.
VX.
VIL
VIII.
EL
X.
Total.
1
2
2
2
1
z
I
2
2'
z
z
z
10
S
5
3
5
6
5
6
4
4
2
z
41
4
1%
10
11
7
«4
9
8
IS
zz
zz
109
6
«3
16
«9
20
12
»3
zz
17
14
H
149
6
*3
SI
»9
13
20
»S
19
2Z
22
16
199
7
20
>9
sz
18
20
20
»3
z8
«3
22
194 '
8
lO
18
7
14
»3
zo
15
9
17
17
130
9
6
8
8
8
13
7
zo
5
5
10
80
10
6
4
6
3
2
2
7
5
5
5
45
11
3
I
2
8
4
z
2
6
2
29
12
I
2
z
3
z
z
9
13
I
z
z
3
Number 1
of primes J
658
655
63,
678
634
645
669
636
669
679
6554
Number of oeuturiee each of which oontaina n primes.
n.
1
1
-
I.
n.
Ml.
IV.
V.
VL
VIL VIII. L2L
X.
Total
1
z
z
z
z
4
2
2
4
2
2
z
3
z
3
2
z
21
3
5
3
3
3
6
5
5
3
z
5
39
4
6
8
9
8
7
9
6
7
zo
13
83
5
z8
13
z8
H
20
z6
Z2
z6
2Z
Z2
160
6
18
2Z
2Z
20
20
«4
2Z
«9
z8
z8
200
7
15
»5
H
•26
IZ
zo
24
H
»9
26
194
8
18
«7
16
z6
z8
'5
9
8
8
Z2
137
9
12
3
9
5
7
7
9
»3
14
7
86
10
3
2
7
2
4
5
9
3
6
5
46
11
2
z
z
3
3
5
3
3
z
22
12
'
2
z
2
z
z
7
13
z
1
Number "1
of primee J
651
634
<J53
655
650
640
683
66z
654 64Z
6522
6,000,000 to 6,100,000.
6,100,000 to 6,200,000.
Number of centuries each of which contains n primes.
I.
n.
111.
IV.
V.
VI.
VTT.
Vlll.
TX.
X.
Total.
1
I
z
3
2
2
2
2
2
z
3
14
3
5
2
5
3
4
6
5
4
5
42
4
7
zz
zo
14
*
zo
zz
13
Z2
12
107
5
2Z
22
13
15
25
14
17
zz
14
»3
165
6
18
20
22
«9
22
21
H
z8
H
20
208
7
12
18
14
18
14
17
H
a3
z6
»9
165
8
18
Z2
z6
»3
14
«4
12
9
13
15
136
9
ZZ
3
zo
12
9
10
7
10
9
86
10
4
5
2
2
5
3
6
4
3
38
11
z
4
6
2
4
2
4
z
3
29
12
z
z
z
z
2
7
Number 1
of primes/
639
639
658
638
628
668
639
648
6SS
646
6458
Number of centuries each of which contains
n primes.
n.
L
n.
TTI.
IV.
V.
VI.
VU.
VIII.
TX.
X.
Total
1
1
1
z
p
z
z
3
2
z
2
z
z
2
z
z
z
z
11
3
6
2
4
2
7
3
3
2
z
z
31
4
12
13
15
13
zz
Z2
»3
z6
zz
z8
134
5
z8
«9
z6
»S
II
17
»3
»3
2Z
18
161
6
20
22
IZ
20
»5
20
19
>9
»3
«5
194
7
Z2
8
12
16
20
14
20
a3
z8
«5
158
8
Z2
26
«7
16
12
19
20
6
n
17
158
9
IZ
8
«3
6
6
5
8
zz
4
7
79
10
4
2
6
10
4
6
2
6
6
3
49
11
2
2
z
z
z
z
z
z
3
13
12
z
z
I
2
5
13
I
2
3
Number 1
of primee j
632
641
655
660
639
644
645
63Z
641
648
6436
46
INTBODTJCTnON.
Kii.
5,200,000 to 6,300,000.
5,300,000 to 5,400,000.
M^
Number of centurief m/ch. of whioh oontuas n primee.
9»
L
n.
m.
IV.
V.
VI.
vn.
VI IL
rx.
X.
TotaL
o
1
o
X
2
3
2
o
I
I
I
X
X
X
X
8
3
3
6
4
5
4
6
3
2
8
46
4
XI
6
5
14
XX
«5
10
9
13
97
5
x8
18
x6
9
H
17
14
x8
xo
22
156
6
aft
11
30
IX
18
20
19
2X
22
H
218
7
x6
x6
ftO
19
17
10
M
22
22
17
183
8
1%
«3
13
20
16
IS
XX
X2
XO
14
136
9
lO
xo
%
9
9
9
6
xo
7
79
10
5
6
6
2
4
4
8
4
5
47
11
o
%
2
X
4
2
2
2
17
12
3
X
X
X
7
13
X
X
3
Number 1
of primeej
659
649 650
647
673
630
649
653
629
654
^493
Number of oeatones each of whioh oontains n primes.
Ha
L
n.
m.
IV.
V.
VL VU. VIU.
IX
X.
TotaL
1
X
1
2
3
X
2
X
9
3
4
5
9
9
6
2
3
5
X
48
4
10
14
9
8
XX
14
«4
12
10
17
119
6
H
16
x6
16
13
22
x6
13
21
13
160
6
22
18
20
«4
2X
15
15
x8
X2
20
175
7
22
13
«5
x6
«4
20
26
22
2X
22
191
8
x6
XX
17
2X
14
X2
13
X3
9
13
139
9
5
xo
7
8
14
9
9
14
7
91
10
5
7
5
5
2
5
4
3
4
44
11
X
3
2
X
3
2
14
12
X
X
2
X
7
13
X
1
14
X
1
Number 1
of primes j
654
637
643
649
648
642
642
«53
654 640
6461
5,400,000 to 6,500,000.
5,500,000 to 6,600,000.
«a
Number of centuries each of whioh oontains n primes.
II.
L
IL
m.
rv.
V.
VI.
VIL
VIII.
rx.
X.
Total.
1
X
2
2
X
6
2
X
X
2
2
2
X
X
10
3
6
6
7
2
3
4
xo
4
4
4
60
4
XX
xo
5
7
6
13
4
10
8
14
88
6
15
14
»3
«9
23
XX
17
x8
x6
13
159
6
22
20
20
as
22
22
17
22
21
21
212
7
H
»3
19
17
x8
>9
18
*S
22
>9
194
8
15
5
20
13
13
xo
20
X2
XO
XX
129
9
XX
XX
8
8
7
8
xo
6
»5
9
93
10
3
3
4
S
6
6
2
X
3
8
41
11
X
3
2
2
2
X
X
12
12
X
2
X
4
13
X
X
2
Number!
of primes/
630
640
657
650
63s
659
641
623
6Si
652
6438
Number of oenturies each of which oontains n primes.
II.
1
1
L
n.
ni.
IV.
V.
VI.
Vli. VIII. IX.
X.
Total.
X
1
1
2
X
8
2
2
X
X
X
X
4
2
2
15
3
4
4
7
3
S
3
6
S
40
4
7
14
12
7
14
13
10
8
9
108
6
»S
9
»S
19
18
21
10
17
x6
166
6
«3
28
»s
x8
x8
16
17
20
22
198
7
22
14
»s
23
21
19
x6
21
»3
190
8
'S
X4
6
xo
IS
13
20
14
12
136
9
8
X2
IX
4
8
9
7
12
83
10
4
3
I
5
S
4
4
35
11
3
X
4
2
I
2
X
17
12
X
X
X
I
X
X
7
13
X
•
1
Number 1
of primes
638
623
633
667
618
634
65s
636
641
657
6402
$n.]
ENUMEBATION OF FBIMES IN THE FIBST NINE MILLIONS.
47
6,600,000 to 5,700,000.
5,700,000 to 5,800,000.
aa
Nunber of oenturiee eaeh of which oontaixiB n primes.
n*
I.
n.
in.
rv.
V.
YI.
Vll.
vm.
IX.
X.
Total
o
1
1
I
3
2
%
I
I
I
I
1
1
10
3
6
8
6
6
6
3
4
s
1
3
49
4
8
8
II
10
II
10
7
14
II
9
99
6
«7
10
10
17
11
14
10
13
18
14
155
6
19
33
IS
IS
»7
18
as
16
16
14
218
7
11
>S
17
16
14
17
18
10
IS
10
184
8
17
IS
IS
13
10
14
13
16
13
11
148
9
1
3
9
7
13
10
8
9
6
9
76
10
5
6
3
S
3
1
3
3
5
6
41
11
a
1
1
I
I
I
I
I
1
1
15
12
o
I
I
2
Nmnber 1
of primee
633
638
638
643
624
641
63s
638
635
679
6404
Number of oenturiee each of which oontains n primes.
n.
I.
n.
m.
IV.
V.
VL
Vll.
VIII.
DL
X
Total
1
I
3
2
I
1
4
ft
I
18
3
6
9
6
9
8
5
ft
G9
4
11
IS
6
II
II
ift
II
99
6
IS
15
18
9
10
11
13
IS
15
19
151
6
10
16
*s
18
19
11
19
17
ftft
fti
189
7
H
11
19
11
17
H
11
»s
ftO
*4
216
8
10
13
14
II
17
11
9
II
9
114
9
7
18
S
11
8
4
7
s
81
10
3
4
6
4
6
5
6
47
11
3
ft
1
I
3
ft
15
12
1
I
6
13
I
2
Number 1
of primee j
63s
667
631
632
6S4
636
617
637
637
640
6387
6,800,000 to 5,900,000.
6,900,000 to 6,000,000.
Number of centuries each of which contains 11 primes.
n.
1
I.
n.
III.
IV.
V.
VI.
vn.
VIII.
IX.
X.
Total
I
I
2
1
I
ft
I
4
2
I
ft
ft
ft
I
ft
10
3
6
3
8
s
3
10
7
4
4
50
4
8
9
13
6
10
Ift
14
8
6
8
94
5
IS
^3
17
21
17
20
II
18
11
18
172
6
44
IS
*9
II
16
20
16
16
*3
fti
191
7
IS
»3
ftO
18
»3
17
17
ftO
as
22
200
8
IS
13
II
12
9
12
II
IS
ID
IS
123
9
s
7
S
6
8
13
6
13
10
79
ID
ID
S
ft
7
6
s
S
6
I
52
11
ft
3
4
2
ft
I
I
15
12
ft
I
I
5
13
I
I
2
14
1
Number 1
of primee
654
63ft
617
641
6S7
627
63 s
641
677
645
6436
n.
Number of centuries each of which oontains • primes.
I.
n.
TIT.
IV.
V.
VI.
VIL
vin.
iX.
X.
Total.
1
ft
2
2
ft
ft
3
I
I
I
ft
ft
16
3
7
ft
S
9
8
ft
6
6
4
55
4
8
Ift
8
7
10
13
II
IS
16
107
5
13
IS
19
20
19
14
14
IS
17
6
152
6
fti
IS
18
16
16
22
*9
14
ft3
*3
197
7
19
fti
17
19
21
*4
18
IS
19
17
190
8
IS
II
9
10
8
II
9
ftO
13
18
124
9
7
14
Ift
6
9
9
6
6
9
81
10
4
5
S
7
6
4
4
4
ft
47
11
4
3
4
4
I
ft
I
21
12
I
ft
2
6
13
1
14
I
1
Number 1
of primes j
648
667
649
646
633
639
633
636
644
62s
6420
48
INTEODUCTION.
Wn;
6,000,000 to 6,100,000,
Nnmber of oantories each of which oantains
11 primes.
At
1
I.
n.
m.
IV.
V.
VI.
VII. vrn.
IX.
X
TotaL
I
I
2
1
o
I
I
2
2
3
2
2
2
I
3
3
17
3
4
6
3
5
7
5
7
6
4
52
4
II
9
9
14
12
10
12
13
12
H
116
6
11
1%
H
«5
14
16
20
13
»3
II
140
6
»5
«7
20
*9
19
20
21
*7
16
»5
189
7
H
»9
21
19
20
12
14
*4
21
20
204
8
»3
II
10
7
12
19
17
12
II
18
130
9
10
6
II
4
10
8
4
II
II
79
10
4
5
5
3
I
7
5
5
2
43
11
3
I
I
2
2
3
I
2
18
12
I
2
6
13
o
o
I
I
2
14
I
o
1
Number 1
of primes/
66 1
638
664
601
627
666
61S
653
628
644
6397
6,100,000 to 6,200,000.
^m
Number of oentiiriee each of which oontaias n primes.
!••
I.
n.
Ul.
IV.
V.
VL VII.
VIII.
DC
X
TofaL
1
I
1
2
%
3
I
I
2
2
I
2
3
2
19
3
4
3
5
8
5
6
6
5
6
9
67
4
8
12
12
10
10
12
7
10
18
10
109
5
15
II
16
13
»5
»S
12
16
»3
II
147
6
20
26
20
«9
12
16
>9
45
16
»4
187
7
20
17
18
22
15
20
*3
20
16
21
192
8
15
14
14
12
»5
8
»5
8
14
17
132
9
12
9
9
9
10
13
8
8
8
10
96
10
3
3
3
5
2
5
6
5
4
2
88
11
I
2
I
2
3
2
2
13
12
I
2
I
I
I
6
13
I
I
2
14
I
1
Number 1
of primeB J
652
641
636
637
637
645
658
627
619
650
6402
6,200,000 to 6,300,000.
6,300,000 to 6,400,000.
^a
Number of oenturiee each of which oontaiiiB n primes.
n.
I.
n.
III.
TV.
V.
VI.
VII.
VIII.
IX
X
Total.
1
2
I
4
2
3
I
4
3
3
I
I
I
20
3
3
5
2
3
6
5
5
3
5
37
4
II
13
12
12
9
9
12
>3
12
8
111
6
15
19
IS
13
19
13
«7
18
«7
«7
163
6
»4
18
17
44
44
18
10
14
16
17
182
7
25
21
14
19
15
21
»5
16
»9
>5
190
8
10
20
«5
H
19
12
21
12
20
148
9
7
10
6
4
9
4
8
9
13
76
10
4
3
2
6
2
7
3
8
2
44
11
3
2
2
2
4
I
2
2
22
12
I
I
I
I
4
yvaaber ^
of primes J
644
602
653
632
637
650
647
636
664
660
6425
1
2
3
4
5
6
7
8
9
10
11
12
13
Number of centuries each of which contains n primes.
I.
Number 1
of primes/
o
5
12
17
13
»4
19
5
2
I
2
646
n.
I
I
12
«9
21
22
»S
3
I
I
2
m.
2
7
5
IS
22
18
«4
13
2
IV.
I
2
12
13
as
18
18
8
2
V.
VI.
vn.
vra.
2
6
II
19
14
17
17
II
2
634
649
633
630
2
6
13
16
16
«7
12
10
7
3
3
9
44
20
17
12
3
6
3
o
4
S
IS
14
IS
17
13
9
5
I
3
6
xo
IS
23
16
12
10
4
o
t
I
II
17
22
21
13
12
2
o
637
626
609
628
TotaL
19
42
110
169
191
187
145
84
33
9
64s
6337
§11.]
ENUMERATION OP PEIMES IN THE FIEST NINE MILLIONS.
49
6,400,000 to 6,500,000.
6,500,000 to 6,600,000.
^
Number of oentuiies eadh of which contains n primes.
Ik
I.
n.
ni.
IV.
V.
VI.
VU.
vm.
TX.
X.
Total
o
1
I
I
I
I
4
2
2
4
4
5
I
2
I
I
I
21
3
3
3
5
5
8
2
3
4
7
2
42
4
II
10
H
10
10
II
12
12
II
II
112
5
»5
13
»4
19
18
16
19
19
II
20
174
6
12
18
»5
H
13
20
14
21
17
18
172
7
17
as
14
II
20
21
26
21
23
17
195
8
»5
14
»7
12
14
18
II
II
16
»9
147
9
8
8
12
8
9
6
9
5
8
6
79
10
2
2
2
4
2
3
3
3
4
3
28
11
4
2
2
2
3
2
I
2
I
I
20
12
o
I
I
I
I
I
5
13
o
I
1
Number 1
of primeBj
630
643
635
611
633
659
632
628
637
639
6347
Number of centuries eadh of which contains n primes.
fit
I.
II.
111.
IV.
V.
VI.
Vll.
VllL IX.
X.
Total
1
I
r
2
I
4
2
I
2
3
2
I
16
3
5
5
4
9
5
3
6
6
3
4
50
4
«S
«3
6
6
4
7
12
6
12
II
92
5
H
14
13
H
19
22
17
16
II
17
157
B
«5
22
M
23
as
a3
16
'S
20
21
204
7
26
13
»3
»7
22
16
22
a7
»3
19
208
8
II
19
II
17
13
8
16
14
10
18
137
9
9
7
9
II
8
10
7
10
12
S
88
10
2
4
4
I
4
2
3
4
3
27
11
2
I
I
I
4
2
2
I
14
12
I
I
I
I
I
5
13
c
14
I
1
Number "1
of primes j
642
640
646
629
641
640 634
639
659
632
6402
6,600,000 to 6,700,000.
6,700,000 to 6,800,000.
AM
Number of centuries each of which contains n primes.
n*
L
n.
III.
IV.
V.
VI.
VII.
vm.
IX.
X.
Total.
1
I
1
2
s
4
2
I
2
I
I
3
19
3
6
3
S
7
6
3
3
5
3
3
44
4
13
7
9
12
9
14
4
12
17
12
109
5
IS
16
IS
13
18
as
18
IS
13
20
168
6
18
17
22
26
20
17
28
21
20
17
206
7
18
17
17
16
IS
18
19
as
20
17
182
8
IS
IS
IS
9
13
6
17
8
12
13
123
9
10
12
II
8
IS
9
6
9
8
12
100
10
3
3
I
7
3
4
4
I
2
S
33
11
2
4
2
2
2
I
13
12
I
•
I
2
Number 1
of primes j
641
662
615
620
640
6,3
648
630
618
651
6338
Number of centuries each of which contains n primes.
91.
L
n.
in.
IV.
V.
VI.
VII.
VUl.
IX.
X.
Total
1
1
1
I
I
2
4
2
I
I
3
I
3
6
15
3
4
7
4
2
4
3
6
4
9
2
45
4
7
8
13
II
8
12
13
9
13
8
102
5
17
17
8
14
21
6
14
19
13
18
147
6
19
a4
a7
a3
20
18
28
22
16
20
217
7
20
20
17
22
22
13
17
as
22
19
197
8
17
8
16
16
IS
26
10
14
13
13
148
9
8
7
9
4
3
12
8
2
9
6
68
10
3
6
4
3
2
4
4
4
3
6
39
11
2
I
I
2
I
I
I
2
11
12
I
I
I
3
13
I
c
2
3
Number 1
of primes
^SS
634
646
634
641
661
619
632
620
633
637s
H
I
60
INTEODUCnON.
[§n.
6,800,000 to 6,000,000.
Mm
Number of oenturiee eaoh of wbich contains n primes.
It*
L
n.
fll.
rv.
V.
VI. vu. vm.
IX
X.
Total.
o
1
I
2
I
4
2
I
3
2
3
1
I
3
16
»
6
3
II
7
4
5
6
4
5
6
67
4
8
13
9
9
7
4
9
12
10
8
89
5
17
20
16
»s
22
14
18
14
>5
18
169
6
28
18
18
21
H
as
18
«9
20
«7
206
7
12
»s
18
17
16
16
19
44
20
20
177
&
14
»3
14
14
9
10
9
II
•
10
12
116
9
4
10
3
6
10
17
II
6
14
9
90
10
5
5
9
6
5
5
6
3
2
3
49
11
4
I
2
2
I
I
4
3
3
I
22
12
2
2
4
Number ^
sf prixneB j
632
653
634
631
633
663
660
623
651
631
6411
6,900,000 to 7,000,000.
n.
Number of oeninriee each of which contains • primes. 1
I.
n.
TIT.
IV.
V.
VL VU.
VIII
UL
X
TotaL
I
1
1
I
I
a
I
5
2
t
2
2
I
4
%
12
3
5
3
5
4
8
4
8
3
9
66
4
10
II
7
12
8
12
7
10
9
99
6
«3
16
16
16
15
18
«5
22
«S
169
6
28
21
20
'5
19
H
18
20
22
192
7
>9
*3
22
22
19
10
18
16
18
184
8
10
13
13
»4
16
H
'4
22
10
140
9
7
7
6
9
9
II
II
2
12
81
10
5
3
7
5
3
8
3
I
40
11
2
2
I
I
4
2
14
12
I
I
2
I
7
Number 1
of primeaj
637
641
615
652
642
631
648
632
630
637
6365
7,000,000 to 7,100,000.
7A00,000 to 7,200,000.
MM
Number of oenturies each of which contains
n primes.
»•■
I.
n.
lU.
IV.
V.
VI.
Vll.
VIM.
TX.
X
Total.
1
I
I
2
2
4
I
I
I
2
3
4
I
I
18
8
3
9
4
6
2
II
8
8
4
6
61
4
9
6
9
t
«5
7
16
7
8
II
96
6
15
17
20
18
II
>3
ID
12
iS
16
160
6
*3
18
20
»3
18
16
22
12
as
19
196
7
18
H
14
13
H
22
20
20
17
a3
i95
8
«4
»3
18
13
13
16
8
21
16
II
143
9
8
10
10
10
10
II
8
5
9
86
10
3
I
5
4
I
3
4
3
31
11
2
2
4
3
I
I
I
2
I
20
12
1
13
ft
I
1
Number 1
lof primes j
629
630
654
648
657
628
594
660
640
629
6369
fi.
Number of oenturies each of which contains
n primes.
I.
n.
ni.
IV.
V.
VI.
VII. VIII.
IX
X
Total.
I
1
1
I
2
I
4
2
I
I
2
3
4
1
4
2
18
3
4
8
7
6
4
6
6
6
7
1
55
4
9
10
II
14
10
9
IS
13
10
IS
116
6
16
20
14
14
14
H
14
18
20
IS
159
6
as
'9
20
»9
a4
18
16
17
19
H
201
7
«S
21
as
17
8
17
16
14
II
19
163
8
9
»3
9
12
19
IS
19
16
17
15
144
»
IS
S
7
ID
12
10
7
7
6
8
85
10
s
4
2
I
3
6
2
2
4
2
31
n
I
I
I
I
I
I
I
2
4
I
14
12
I
1
•
I
I
2
6
13
I
1
14
I
1
15
I
1
Number 1
of primes j
663
609
611
617
6S4
640
629
61S
630
638
6306
§11.]
ENUMERATION OF PEIMES IN THE PEBST NINE MILLIONS.
51
7,200,000 to 7,300,000.
7,300,000 to 7,400,000.
Number of oentuzies each of which oontains n primee.
It.
1
L
n.
TTI.
IV.
V.
VL
vu. vni.
n.
X
Total
o
1
o
I
2
2
2
I
2
6
I
2
2
19
8
7
8
9
6
2
8
5
5
7
58
4
9
7
6
II
10
«3
8
12
9
92
5
15
16
20
»3
13
12
'7
»3
II
17
167
6
21
13
21
10
»3
»5
»4
H
22
20
193
7
20
22
«S
16
21
17
20
20
21
17
189
8
«4
10
17
20
18
14
12
15
15
13
148
9
6
8
6
9
9
8
7
8
7
9
77
10
4
3
3
2
2
4
7
5
3
4
37
11
I
2
2
I
I
2
I
2
»
14
12
I
I
I
3
13
I
1
Number "1
of primeej
632
630
636
625
653
615
647
642
636
632
6348
Number of centuries eaoh of which oontaine n primee.
Hi.
I.
n.
ITT.
IV.
V.
VI. VU.
VIII. IS.
X.
TotaL
1
I
1
2
3
4
2
3
2
3
4
3
24
3
4
6
4
4
8
4
5
6
5
4
50
4
II
9
»3
15
II
6
16
7
>3
9
110
5
18
19
»9
20
18
»3
«3
17
9
17
173
6
20
18
21
»7
«7
20
«7
20
21
28
199
7
13
10
14
»7
17
14
17
19
22
12
155
8
X6
»9
18
14
10
17
18
13
12
18
155
9
7
xo
5
6
8
9
8
10
8
6
77
10
4
2
3
4
6
4
4
4
6
3
40
11
3
3
»
X
2
I
12
12
I
1
13
I
I
I
8
Number \
of primes/
628
636
637
616
621
642
643
6J5
628
619
6299
7,400,000 to 7,500,000.
7^0,000 to 7,600,000.
n.
Number of centuries each of which contains
n primes.
1
I.
n.
in.
IV.
V.
VI.
vn.
VIII.
IX.
X.
TotaL
1
2
3
1
2
3
3
3
X
2
18
3
3
6
6
6
5
9
12
2
3
10
62
4
9
12
12
8
9
9
10
13
13
n
108
5
28
18
"5
«9
«7
16
14
14
16
15
172
6
»9
15
22
26
19
22
21
20
21
20
205
7
14
«9
18
«4
14
13
15
»7
21
18
163
8
8
16
»5
14
17
14
II
18
12
7
132
9
9
5
8
9
9
II
4
6
10
7
78
10
5
5
2
X
5
I
7
5
3
4
38
11
2
2
2
X
2
3
3
15
12
2
X
I
I
5
13
X
2
3
14
I
1
Number 1
of primes J
620
641
633
636
656
615
610
647
632
6x1
6301
Number of centuries eaoh of which contains n primes.
n*
1
I.
n.
m.
IV.
V.
VI.
VII. vrn.
EL
X.
Total.
I
1
1
2
I
2
5
2
2
4
X
2
2
I
2
I
17
3
6
7
3
6
3
5
5
8
4
50
4
x6
»5
14
6
7
»5
12
II
8
109
5
17
19
«7
II
22
20
16
18
16
»5
171
6
16
10
21
29
21
M
«9
»3
19
20
202
7
14
20
18
18
>7
20
17
16
x6
«9
175
8
XX
»9
XI
«5
15
5
II
17
10
16
ISO
9
9
7
7
5
II
9
5
7
II
80
10
7
3
5
7
3
I
9
3
41
11
2
I
2
2
I
I
X
I
I
14
12
2
3
13
I
•
1
14
15
X
1
Number \
of primes /
626
626
6x3
638
653
64s
61S
608
637
644
6305
h2
62
INTEODUCTTION.
[§ n..
7,600,000 to 7,700,000.
7,700,000 to 7,800,000.
Number of oenturiee each of whioh oontaini n primes.
ft.
I.
n.
III.
IV.
V.
VL
VII.
vm.
IX.
X.
TotaL
•
1
1
I
3
2
1
I
5
I
1
I
12
3
3
7
4
4
3
3
7
4
4
44
4
8
16
10
II
8
17
11
8
8
114
5
ao
ts
«S
17
19
14
15
»3
16
172
11
»s
iz
%%
19
11
11
17
11
197
7
17
2S
18
18
19
16
16
X9
11
186
8
«7
I»
16
14
«S
«4
II
14
«9
144
9
8
8
10
6
7
10
10
5
7
75
10
4
I
4
4
I
I
4
4
I
29
U
1
3
I
I
a
1
1
3
17
12
I
1
'%
I
I
7
Number 1
of primes J
657
617
649
611
641
631
616
630
640
643
6347
«.
Number of oentories each of which oontains n primes.
I.
IL
III.
rv.
V.
VI.
Vll.
VUL
IX.
X.
TotaL
1
I
I
I
I
3
7
2
1
3
4
5
I
1
3
4
2
26
3
5
6
4
7
9
5
5
3
6
53
4
9
XI
8
5
II
15
9
13
7
96
5
*3
II
12
11
«5
19
16
20
21
196
6
«9
16
16
16
17
19
25
20
26
190
7
17
10
II
16
15
10
.*3
18
16
II
167
8
14
II
1 1
17
7
II
II
14
II
15
122
9
7
11
9
9
9
5
5
10
3
80
10
I
4
1
9
5
4
I
4
40
11
I
3
I
I
I
2
2
18
12
I
I
3
5
Number 1
of primes j
611
64s
630
607
621
616
612
647
611
633
6245
7,800,000 to 7,900,000.
7,000,000 to 8,000,000.
Number of centuries each of which contains n primes.
1*.
II.
III.
IV.
V.
VI.
Vll.
VIU.
IX.
X.
Total.
I
2
I
I
1
2
3
t
1
I
t
I
I
11
»
S
4
5
3
10
a
s
7
47
4
13
IS
10
8
10
«3
la
11
120
5
S A
14
16
13
IS
14
14
17
156
e
10
ao^
17
a6
30
*3
19
IS
194
7
aa
II
19
16
19
*$
aa
11
190
8
17
H
16
«7
la
16
la
IS
142
9
< 10
1
«3
6
7
S
S
10
8
84
10
*
: I
1
4
a
3
3
a
4
3
34
11
i
• a
1
1
I
3
I
3
I
14
13 '
1
1
I
I
I
5
•
Number '
of primes/
63S 633 614
641
647
631
634
647
641
626 1
«3«*
Number of centuries each of which contsins
A primes.
•.
I.
n.
IIL
IV.
V.
VL
Vll.
Vill.
DL
X.
TotoL
1
3
I
I
6
2
I
I
a
1
1
8
8
6
6
3
3
4
S
9
8
11
6
61
4
10
13
II
13
8
II
8
It
II
9
105
5
x6
17
13
10
16
*7
11
15
16
14
175
6
aa
H
a4
18
a4
11
19
«4
10
30
216
7
19
16
19
11
18
11
14
ai
13
17
171
8
la
10
19
13
11
14
x6
11
13
13
134
9
II
9
8
9
5
$
4
3
S
6
65
10
3
a
I
8
3
5
8
6
3
39
11
•
a
I
I
1
4
4
3
17
12
I
I
a
4
Number *)
of primes/
630
1
616
634 619
640
604
634
638
634
615
6274
§n.]
ENUMEEATION OF PEIMES IN THE FIBST NINE MILLIONS.
»
8,000,000 to 8,100,000. 8,100,000 to 8,200,000.
53
».
Number of centuries each of which contains n primes.
I.
n.
ni.
IV.
V.
VI.
Vll.
VIII.
IX.
X.
Tptal.
o
I
1
1
I
1
2
4
I
2
I
0.
3
2
2
16
3
5
6
•
9
5
8
6
2
7
9
4
61
4
12
7
3
17
13
8
14
10
10
17
111
6
13
26
19
18
10
16
18
as
22
22
189
6
15
22
24
»4
19
26
19
12
17
17
195
7
17
18
18
II
20
16
20
17
H
15
166
8
13
12
7
17
16
II
II
»s
17
II
130
9
«5
5
7
5
7
8
5
5
6
7
70
10
3
2
6
3
3
7
5
3
4
I
37
11
2
4
I
I
3
2
3
16
12
O
I
I
I
I
I
6
13
o
1
I
I
3
M umber
of primes J
637
609
631
605
639
652
628
623
619
607
6250
Number of centuries each of which contains n primes.
ft.
I.
11.
in.
IV.
V.
VI.
VU.
VIU.
IX.
X
Total.
o
1
o
I
I
I
3
2
I
3
2
5
2
I
2
2
2
20
3
3
3
5
6
8
4
8
4
7
7
55
4
7
9
10
10
10
9
7
14
9
II
96
5
28
19
17
16
»3
t6
18
«S
15
19
176
6
^3
22
17
44
13
22
21
20
20
20
212
7
14
14
18
19
17
18
«S
18
14
II
168
8
16
15
10
10
16
20
II
13
8
17
136
9
5
14
5
6
5
9
7
9
9
76
10
2
6
3
5
2
4
4
6
4
41
11
I
2
I
I
I
z
8
12
2
2
I
6
13
I
I
I
3
Number 1
of primes j
619
633
651
602
622
64.8
63s
625
648
618
6301
8,200,000 to 8,300,000.
8,300,000 to 8,400,000.
Mm
Number of centuries each of which contains
n primes.
**.
L
11.
III.
IV.
V.
VI.
Vll.
VIM.
IX.
X.
Total.
1
I
I
I
I
I
5
2
2
i
I
2
I
3
I
3
2
16
3
8
5
4
4
3
6
4
4
8
6
52
4
9
10
18
10
9
12
12
13
»3
14
120
5
19
«7
10
20
20
18
«3
17
18
10
162
6
20
»9
28
18
23
15
^7
»9
II
18
198
7
14
20
14
23
17
23
20
20
15
^3
189
8
12
II
13
II
II
»9
7
9
*3
18
124
9
7
9
5
8
8
3
10
II
8
3
72
10
4
4
6
3
4
3
4
3
8
2
41
11
6
2
I
I
2
I
3
16
12
I
2
3
13
I
1
14
I
1
Number 1
of primes j
639
630
623
638
631
618
619
629
634
622
6283
Number of centuries each of which con
tains
n primes.
fl«
I.
11.
ni.
IV.
V.
VI.
VII.
vm.
IX.
X.
Total.
I
1
1
2
z
3
2
z
3
I
2
2
3
3
z
2
18
3
4
5
8
10
8
6
3
5
7
3
59
4
»3
9
9
7
4
10
12
10
6
8
88
6
15
20
17
»4
21
16
17
20
26
21
197
6
13
19
19
14
18
20
17
»9
21
21
191
7
20
16
17
17
22
22
30
22
iz
as
202
8
12
9
»3
II
8
II
9
8
10
7
98
9
5
8
9
8
9
9
5
6
7
8
74
10
6
4
4
4
5
3
2
4
8
4
44
11
I
6
I
3
4
I
2
3
21
12
I
z
2
13
I
1
14
I
1
Number "i
of primes/
631
646
618
618
658
627
608
623
636
620
628s
64
INTEODITCTION.
Kn.
8,400,000 to 8,600,000.
8,600,000 to 8,600,000.
Number of oenturies mch of which oontaixui » primes.
ft.
1
I.
n.
111.
IV,
V.
VI.
VII. vm.
IX.
X.
Total.
I
1
1
2
2
4
2
5
2
2
2
2
I
2
2
19
3
4
8
10
8
9
4
8
2
4
63
4
9
9
10
8
7
II
12
8
II
8
93
5
i8
18
16
18
20
20
18
12
22
20
182
6
20
z6
18
18
z8
19
25
26
16
»S
201
7
ax
«5
15
«9
21
17
'7
22
12
18
177
8
zo
14
15
19
II
II
10
II
15
12
128
9
9
8
4
5
10
II
10
4
76
10
3
6
4
2
5
2
5
5
39
11
I
4
3
2
3
2
18
Number
of primes J
617
643
611
619
602
641
612
633
639
628
6245
mm
Number of oeatariee each of whidi eontains n primes. 1
n.
I.
n.
III.
IV.
V.
VL
Vll.
VI IL
IX.
X.
TotaL
1
I
I
X
3
2
I
3
2
I
3
1
I
I
6
21
3
3
4
5
5
4
2
4
5
6
43
4
8
II
ID
16
'3
16
10
14
7
120
5
22
22
18
13
22
17
10
15
IS
172
6
22
14
18
21
16
15
29
18
z6
182
7
18
18
17
19
14
20.
21
20
22
188
8
13
la
II
14
II
13
ID
16
II
126
9
7
9
II
5
8
9
8
10
10
86
10
4
5
4
4
6
3
I
4
38
11
I
I
I
2
1
2
2
12
12
I
I
I
2
I
7
13
1
2
Nmnber 1
of primes j
641
632
641
625
621
643
621
654
624 624
6326
8,600,000 to 8,700,000.
8,700,000 to 8,800,000.
Number of oenturies each of which contains n primes.
n.
I.
n.
TTT.
IV.
V.
VI.
Vll.
VIU.
TX.
X.
Total
1
I
I
2
2
I
I
3
2
3
2
3
16
3
10
6
6
7
6
5
7
6
6
69
4
10
5
16
H
13
8
9
13
9
II
108
5
»3
18
14
9
»9
»9
>5
28
20
20
166
6
45
20
18
21
17
19
19
16
14
21
190
7
17
21
13
19
19
14
21
10
19
14
167
8
7
14
14
17
II
14
18
8
21
17
141
9
>3
6
II
3
8
»3
8
10
3
4
79
10
2
4
2
7
z
2
I
3
5
6
33
11
I
I
2
2
I
4
I
I
I
1
16
12
I
I
I
I
2
6
Number 1
of primes j
642
626
622
637
602
651
629
592
651
629
6281
II.
Number of centuries each of whidi contains » primes.
L
IL
in.
IV.
V.
VI.
VIL
VMl.
iX.
X.
Total
1
I
I
I
3
2
I
3
I
3
2
3
z
5
19
3
7
7
8
II
3
7
5
3
4
8
63
4
12
zo
7
8
13
9
8
17
13
12
109
5
16
13
18
«S
13
II
13
18
12
14
143
6
20
If
22
21
»7
»5
14
17
14
16
197
7
19
18
15
17
18
17
13
22
14
16
189
8
II
17
18
17
8
'5
15
5
17
16
139
9
9
13
7
7
II
9
8
10
9
9
92
10
5
6
3
4
3
4
2
4
3
2
36
11
I
I
I
I
2
2
8
12
z
1
13
I
1
Number 1
of primes /
623
649
629
624
614
637
653
600
659
611
6299
§n.]
ENT7MEBATI0N OF FBIMES IN THE FIBST NINE MILLIONS.
55
8,800,000 to 8,900,000.
8,900,000 to 9,000,000.
ft.
Number of oeutariee each of which oontains n primeB.
I.
n.
III.
IV.
V.
VL
Vll.
VIII.
IX.
X.
Total
1
2
3
4
5
6
7
8
9
10
11
12
O
I
3
5
XI
»9
H
22
12
7
5
O
I
3
2
5
14
«5
ao
II
x6
10
2
2
I
3
3
8
XI
»9
18
10
9
7
I
7
«7
«5
as
15
9
7
3
2
2
7
12
13
22
16
17
4
4
3
4
7
10
20
»9
16
6
•
I
2
I
4
6
9
17
»3
15
14
8
3
I
2
8
II
IS
IS
22
S
10
4
4
1
I
I
10
12
12
20
18
XI
II
3
I
I
4
12
22
17
x6
13
10
2
2
x
6
22
62
116
159
204
167
126
82
34
18
4
Number 1
of piimee j
620
610
645
606
627
607
615
641
615
634
6220
«*
Number of oenturiee each of which oontainB n primes.
w.
I.
n.
m.
IV.
V.
VI.
VII. VIII.
IX.
X.
TotaL
I
1
1
I
I
I
I
4
2
I
2
I
X
I
3
2
2
13
3
I
4
6
4
8
S
8
7
4
6
53
4
16
16
8
12
9
14
14
8
IX
9
117
5
^3
II
*3
19
17
14
17
19
x8
16
177
6
17
17
14
x8
26
as
20
22
19
x8
196
7
17
17
16
20
14
14
18
x8
20
21
175
8
12
12
19
10
15
14
7
XI
13
17
130
9
S
to
8
9
7
S
7
9
6
6
72
10
6
7
4
5
3
4
6
3
S
4
47
11
2
X
I
2
I
3
I
11
12
I
I
I
I
4
Number "I
of primeej
607
647
639
628
614
615
613
620
653
634
6270
In the second volume of Gauss's Werke there are given tables, similar in form to the
preceding tables, for the second and third millions, and which therefore correspond to those on
pp. 35-40 : they were calculated by Goldschmidt, and communicated by him to Gauss. These
tables are very inaccurate, the errors being so numerous that it is not worth while to give a list
of them : there are, for example, no less than sixteen errors in the body of the first table
(1,000,000-1,100,000). The following tables* give a comparison of the values obtained from
these tables for the complete groups of 100,000 with the values found in the present enume-
ration : —
Sscoin) MiLLioir.
limits.
Number of Primes.
GftUBS.
Pp. 36-37.
i,ooo,ooo-X|X 00,000
I , X 00,000- X ,200,000
1,200,000-1,300,000
1,300,000-1,400,000
1,400,000-1,500,000
1,500,000-1,600,000
1 ,600,000-1,700,000
1,700^000-1,800,000
1,800,000-1,900,000
1 ,900,000-2,000,000
1,000,000-2,000,000
7,2x0
7»i94
7,081
7,098
7,028
6,971
7,0x2
6,931
6,9SS
6,902
70,382
7,216
7."S
7,081
7.103
7,028
6,973
7,015
6,932
6,957
6,903
70433
Third Milliok.
Limits.
Number of Primes.
Gauss.
Pp. 38-40.
2,000,000-2, X 00,000
2, X 00,000-2, 200,000
2,200,000-2, 300,000
2,300,000-2,400,000
2,400,000-2,500,000
2, 500,000-2,600,000
2,600,000-2,700,000
2,700,000-2,800,000
2,800,000-2,900,000
2^900,000-3,000,000
2,000,000-3,000,000
6,874
6,857
6,849
6,787
6.766
6,804
6,762
6,714
6r744
6,705
67,862
6,874
6,857
6,849
6,79 «
6,770
6,809
6,765
6,716
6,746
6,708
67;885
« t
Frooeedings of the Cambridge FhiloBophical Soeiety/ toL iiu pp. 51, 52.
56
INTEODTrCnON.
[§n.
Tables for the complete Millions.
The following nine tables are similar in arrangement to the ninety tables on pp. 33-55 ;
each column, except the last, relates to 100,000 numbers, and each table to 1,000,000 numbers.
The numbers in the columns of the tables which relate to groups of 100,000 are the same as
those headed "Total" in the preceding tables. The number of primes at the foot of each
column is the number of primes in the group of 100,000 to which it relates.
The results for the nine millions are collected in a final table on p. 61, in which each
column except the last relates to 1,000,000 numbers, and the last relates to the whole nine
millions. It will be seen that the centuries with eight primes are the most numerous in the first
million, the centuries with seven primes in the second and third millions, and the centuries with
six primes in all the other millions. In the whole nine millions collectively the centuries v^th
seven primes are the most numerous ; but the numbers of six-prime and seven-prime centuries
are very nearly equal. The numbers of five-prime and eight-prime centuries are also nearly
equal.
It is remarkable that there should be a century in the third million containing as many as
seventeen primes, and two centuries in the eighth million containing fifteen primes. These
centuries are those which commence with the numbers 2,704,900, 7,133,800, and 7,587,700.
Of the first hundred thousand numbers 9*6 per cent, are primes, and of the second hundred
thousand 8*4 percent., the percentage of primes gradually diminishing to 6*3 at 9,000,000. The
percentages for the different millions are 7-85, 7-04, 6-79, 6-63, 6-54, 6-43, 6-38, 6-32, and 6-28.
Of the whole nine million numbers almost exactly 6*7 per cent, are primes.
Meissel determined the number of primes in the first ten millions to be 664,580, if unity be
included (see p. 35 of the Introduction to the Fourth Million). Assuming this value, the
number of primes in the tenth million is 62,012, and the percentage for this million is 6'20.
to 1,000,000.
n.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
16
16
17
21
26
Number")
primes. )
*^l
o
o
o
3
6
i8
39
io6
149
i8S
177
124
104
46
17
14
8
2
I
I
9593
8
* ■S ^m^
o
I
o
7
15
47
116
145
184
196
146
78
42
18
3
2
o
o
o
o
Number of centuries each of which contains n primes.
8392
<S
o
I
o
13
24
55
134
173
202
167
131
59
29
7
5
o
o
o
o
o
8013
' *5 • *■
o
o
2
12
40
72
127
172
200
171
III
49
29
II
2
2
o
o
o
o
7863
o
I
4
13
39
91
153
175
189
144
86
^Z
33
7
2
o
o
o
o
o
J^ -ia _•*
7678
o
o
3
20
44
87
142
193
185
153
no
40
16
7
o
o
o
o
o
o
^1
7560
o
o
7
16
43
105
145
208
173
78
47
16
6
I
o
o
o
o
o
7445
^ ^5 ^^
o
o
5
19
53
97
162
193
194
119
79
54
17
6
o ^
I
o
I
o
o
7408
<s%
o
o
4
21
53
114
165
182
176
145
76
44
13
5
2
o
o
p
o
o
7323
5"3S-
o
o
4
16
55
"5
179
218
169
119
64
34
17
9
o
I
o
o
o
o
7224
033
o
3
29
140
372
801
1362
i7<55
1821
1554
1058
592
316
122
32
20
8
3
I
I
78,496
§11.]
ENTJMEEATION OF PBIMES IN THE FIEST NINE MILLIONS.
57
1,000,000 to 2,000,000.
n.
1
2
S
4
5
6
7
8
9
10
11
12
13
14
15
16
Number)
pnmes. J
1^1
4
21
53
"S
170
217
164
125
73
39
12
6
o
o
o
7216
8
~SS
5
23
58
102
170
220
156
136
76
35
10
6
2
o
o
Nnmber of centuries each of which contains n primes.
7225
35
o
2
6
32
63
120
160
214
168
III
73
35
9
5
I
o
I
7081
5 So
9
19
69
119
172
207
162
118
72
33
IS
3
o
7103
> o z>
5
7
19
72
130
181
180
183
98
73
34
16
2
o
o
o
7028
? -So
o
2
10
28
77
124
198
172
150
124
63
29
17
6
o
o
o
6973
pi
I
2
II
29
67
121
159
203
174
129
63
27
9
3
I
I
o
7015
i
>Sc>
5
30
70
152
174
194
146
125
61
26
10
5
o
I
o
6932
S"SS
o
o
10
22
71
135
176
204
160
"5
74
23
10
o
o
o
o
^957
5 -So"
04
5
34
67
135
183
221
149
lOI
63
30
5
3
3
o
6903
5-55
r-r Cf
I
16
72
257
667
1253
1743
2032
I6I2
II82
691
3"
"3
39
7
3
I
70,433
2,000,000 to 3,000,000.
n.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Number)
pnmes.^
r^8*
c(i 01
o
3
10
32
69
119
197
204
157
"5
63
21
8
2
o
o
o
o
6874
Number of centuries each of which contains n primes.
fS
(N Cf
O
2
9
27
70
145
183
201
167
no
5«
18
9
4
3
o
o
o
6857
<5o
C9 04
o
2
9
29
73
138
179
205
168
113
44
30
10
o
o
o
o
o
6849
g5g
cf oT
O
4
10
33
86
135
177
194
157
"3
54
29
7
I
o
o
o
o
6791
Sq"
<N CC
9
37
78
146
193
190
151
102
56
25
7
5
o
o
o
o
6770
<5o
C^* (M*
3
5
35
88
136
193
178
173
88
57
25
13
6
o
o
o
o
6809
^8
Cl C9
I'
4
10
28
70
159
195
201
141
96
54
22
17
I
I
o
o
o
6765
O
2
7
43
93
137
195
188
145
87
67
24
9
2
o
o
o
I
6716
^S<o
of of
o
2
15
30
84
141
179
222
132
109
53
18
7
6
2
o
o
o
6746
SScS
C^ CO
o
2
13
44
64
152
187
214
135
103
58
15
II
I
o
I
o
o
6708
I
<• Q
01 CO
25
97
338
775
1408
1878
1997
1526
1036
558
227
98
28
6
1
o
T
67,885
I
L
68
INTEODFCTION.
KiL
3,000,000 to 4,000,000.
n.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Number)
prunes.^
r^8"
CO CO
o
2
II
37
8i
140
206
183
168
94
54
19
4
I
o
6676
Number of centuiies each of which contains n primes.
8-5:
CO CO
3
13
31
87
146
173
198
169
96
52
19
9
2
6717
f 5tf
CO CO
o
I
16
33
91
139
199
193
140
97
50
31
9
I
o
6691
2S^
CO CO
4
13
50
85
133
194
171
169
102
51
23
5
o
6639
Of CO"
o
3
14
48
89
140
172
216
143
100
43
24
6
2
o
661 1
of of
4
12
42
89
188
192
149
102
35
24
3
3
I
6575
of of
o
2
13
31
75
180
190
178
150
94
55
21
6
5
o
6671
00 00
3
17
43
77
146
209
170
176
83
52
21
3
o
6590
of of
4
12
43
99
144
189
169
157
102
45
25
8
3
o
6624
5 5o'
of '^
4
15
42
89
156
209
179
140
80
60
14
7
2
2
6535
^S
eo
30
136
400
862
1480
1929
1849
1561
950
497
221
60
19
4
66,329
4,000,000 to 5,000,000.
n.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Number)
^ \
prunes. /
^8
3
15
29
92
142
201
215
133
93
45
21
7
3
I
6628
Number of centuries each of which contains n primes.
85
3
17
39
90
156
195
2 00
137
89
48
16
7
3
o
6540
3
17
31
no
151
206
161
166
93
37
17
5
2
6510
^ ^
o
o
16
35
100
143
212
205
133
94
44
15
2
I
o
6511
o 5 o
3
13
37
75
153
207
190
163
96
37
18
5
2
6613
2
18
55
96
132
193
192
155
97
35
19
6
o
o
6493
|5|
■ft •*
3
14
48
83
162
193
187
141
97
42
20
6
2
o
^523
8^58"
3
20
49
105
140
188
191
138
97
47
12
9
I
647s
10
41
109
149
199
194
130
80
45
29
9
3
o
6554
»o
o
4
21
39
83
160
200
194
137
86
46
22
7
I
o
6522
2
26
161
403
943
1488
1994
1929
1433
922
426
189
63
18
3
65*369
§11.]
ENUMERATION OF PRIMES IN THE FIRST NINE MILLIONS.
69
5,000,000 to 6,000,000.
Number of oentuiies each of which contams n primes.
n.
1
2
3
4
6
6
7
8
9
10
11
12
13
14
Numher)
prunes./
g'^8"
lO lO
o
3
14
42
107
165
208
165
136
86
38
29
7
o
6458
8^8
I
3
II
31
134
161
194
158
158
79
49
13
S
3
o
6436
S^
iO
3
8
46
97
156
218
183
136
79
47
17
7
3
o
6493
^^Q-'
10 10
o
I
9
48
119
160
175
191
139
91
44
14
7
I
I
6462
ss^
10 to
o
6
10
SO
88
159
212
194
129
93
41
12
4
2
o
6438
fS©
10 kO
3
15
40
108
166
198
190
136
83
35
17
7
I
6402
kO kO
O
3
10
49
99
155
218
184
148
76
41
IS
2
6404
kO kO
O
3
18
S9
99
iSi
189
216
114
81
47
IS
6
2
o
6387
.So"
kO kO
2
4
10
50
94
172
191
200
123
79
S2
IS
5
2
I
6436
•SsT
to CO
o
2
16
ss
107
IS2
197
190
124
81
47
21
6
I
1
6420
»o
4
31
121
470
1052
IS97
2000
1871
1343
828
441
168
S6
IS
3
64>336
6,000,000 to 7,000,000.
n.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Number)
pmnes. /
CD CD
2
2
17
52
116
140
189
204
130
79
43
18
S
2
I
6397
Number of centuries each of which contains n primes.
8"5
1— •
19
57
109
147
187
192
132
96
38
13
6
2
I
6402
CO
o
4
20
37
III
163
182
190
148
75
44
22
4
o
o
6425
fS
CO
19
42
no
169
191
187
145
84
33
9
5
2
6337
^•'S
CD
O
4
21
42
112
174
172
195
147
79
28
20
5
I
o
6347
3q-
o
I
16
SO
92
157
204
208
137
88
27
14
5
o
I
6402
CO CO
o
I
19
44
109
168
206
182
123
100
33
13
2
o
o
6338
I
4
IS
45
102
147
217
197
148
68
39
II
3
3
o
6375
CO
4
15
57
89
169
208
177
116
90
49
22
4
o
o
641 1
ss^
CO b*
I
5
12
56
99
169
192
184
140
81
40
14
7
o
6365
5'5
6
28
173
482
1049
1603
1948
1916
1366
840
374
156
46
10
63*799
i2
60
INTBODUCTION.
[§II.
7,000,000 to 8,000,000.
n.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
16
Number)
prunes. /
r^8"
o
2
i8
6i
96
ISO
196
195
143
86
31
20
I
I
o
o
6369
Number of centuries each of which contains n primes.
8*5
I
4
18
55
116
159
201
163
144
85
31
14
6
I
I
I
6306
^S^
19
58
92
167
193
189
148
77
37
14
3
I
o
o
6348
ss^
o
I
24
50
no
173
199
155
155
77
40
12
I
3
o
o
6299
SSc^
o
18
62
108
172
205
163
132
78
38
15
5
3
I
6301
'^Q-
I
5
17
50
109
171
202
175
130
80
41
14
3
I
o
I
6305
o
3
12
44
114
172
197
186
144
75
29
17
7
o
o
o
6347
?5tf
o
7
26
53
96
196
190
167
122
80
40
18
5
o
o
o
6245
2
I
II
47
120
156
194
190
142
84
34
14
5
o
o
o
6364
sSs^
tm t^ r* 00
O
5
8
61
105
175
216
171
134
65
39
17
4
o
o
o
6274
^S^
00
4
30
171
541
1066
1691
1993
1754
1394
787
360
155
40
10
2
2
63,158
8,000,000 to 9,000,000.
n.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Number")
prunes. /
r^g-
00 00
IS
61
III
189
19s
166
130
70
37
16
5
3
o
6250
Number of centuries each of which contains n primes.
8* o <-C
GO 00
3
20
55
96
176
212
168
136
76
41
8
6
3
o
6301
jS
00 00
o
5
16
52
120
162
198
189
124
72
41
16
3
I
I
6283
5"3
00 00
3
18
59
SS
197
191
202
98
74
44
21
2
I
I
6285
S3^
00 00
I
4
19
63
93
182
201
177
128
75
39
18
o
o
o
6245
5 5s
00 00
o
3
21
43
120
172
182
188
126
86
3^
12
7
2
o
6326
1^1"
00 00
2
IS
59
108
185
190
167
141
79
33
15
6
o
o
6281
00" 00"
3
19
63
109
143
197
189
139
92
36
8
I
I
o
6299
J^Q-
00 00
o
6
22
62
116
159
204
167
126
82
34
18
4
o
o
6220
iSs^
06^
4
13
53
117
177
196
I7S
130
72
47
II
4
o
o
6270
5*3 rf"
00
4
34
178
570
1078
1742
1966
1788
1278
778
390
143
38
II
2
62,760
§11.]
ENUMEEATION OF PMMES IN THE PIE8T NINE MILLIONS.
61
The final results for the nine millions are given in the following table :-
to 9,000,000.
n.
Number of oentories each of which oontainB n primes.
o5
' of
■f^d"
N ©y
ikB^
« -*
'S^
yfi lO
'5sf
lO o
?3q
<o fc^
^S
t> 00
OCT oT
o5:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
21
26
Number)
primes, i
3
29
140
372
801
1362
1765
1821
1 554
1058
593
316
122
3«
20
8
3
I
78,499
I
16
72
257
667
"53
1743
2032
1612
1 182
691
3"
"3
39
7
3
I
o
o
o
25
97
338
775
1408
1878
1997
1526
1036
558
227
98
28
6
I
o
I
o
o
2
30
136
400
862
1480
1929
1849
I561
950
497
221
60
19
4
o
o
o
o
o
2
26
161
403
943
1488
1994
1929
1433
922
426
189
18
3
o
o
o
o
o
4
31
121
470
1052
1597
2000
1871
1343
828
441
168
56
15
3
o
o
o
o
o
6
28
173
482
1049
1603
1948
X916
1366
840
374
156
46
10
3
o
o
o
o
o
4
30
171
541
1066
1691
1993
1754
1394
787
360
15s
40
10
2
2
o
o
o
o
70,433
67,885
66,329
65.369
64,336
63*799
63,158
4
34
178
570
1078
1742
1966
1788
1278
778
390
143
38
II
2
o
o
o
o
o
62,760
24
223
II38
3601
7864
13063
I68I3
I690I
13334
8877
4795
2162
830
272
62
26
9
4
I
I
602,568
It may be remarked that this table shows that there are twenty-four centuries in the nine
millions which contain no prime, and that the distribution of these centuries among the
millions is :
First million . . . .
, none
Sixth million
Second „ ...
1
Seventh „
Third „ . . . ,
1
Eighth „
Fourth „ . . . .
2
Ninth „
Fifth ,
2
• •
• •
• •
• •
4
6
4
4
The first number of each of these centuries, which consist therefore wholly of composite
numbers, is given in the following list :
1,671,800
2,637,800
3," 7,300
3,933>6oo
4,640,600
4,652,400
5,178,200
5,518,700
5*837,400
5,845,200
6,012,900
6,085,000
6,333,800
6,376,200
6,789,800
6,958,700
7,129,900
7,565,200
7,803,500
7,826,900
8,027,700
8,367,400
8,421,300
8,905,200
62
INTEODXrCTION.
[§II.
Seguencea in the Sisth Million.
The following table contains a list of sequences of 81 and upwards occurring in the sixth
million. It is exactly similar to the corresponding tables for the fourth and fifth millions, which
are given respectively on pp. 46 and 10 of the Introductions to these Millions.
5,000,000 to 6,000,000.
Sequences of 81 and upwards.
Lower limit.
Upper limit.
Sequence.
Lower limit.
Upper limit.
Sequence.
Lower limit.
Upper limit
Sequence.
5,001,809
5,001,907
97
5.377.439
S.377.529
89
5.734.357
5.734,439
81
5»oi4,579
5,014,663
83
5.378,731
5.378,819
87
5.752,403
5.752,489
85
5»oi7>i4i
5.017.231
89
5,386,769
5.386,859
89
5.756,633
5.756,717
83
5,030,821
5.030,917
95
5.392,711
5.392,799
87
5,766,499
5.766,581
81
5,044,673
5,044,769
95
5.398,597
5.398,697
99
5.768,233
5.768,317
83
5>o54,537
5.054.633
95
5.412,133
5.412,217
83
5.774,677
5.774,759
81
5i07?,973
5.072,057
83
5.417.597
5,417,681
83
5.776,483
5.776,579
95
5,074,481
5.074.567
85
5.451.151
5,451,241 .
89
5.779.999
5,780,087
87
5,102,863
5.102,953
89
5.455.643
5.455.727
83
5.783.579
5.783.677
97
5»" 5.947
5,116,039
91
5.459.197
5,459,281
83
5.785.891
5.785,973
81
5»i22,54i
5,122,631
89
5,474,213
5.474,297
83
. 5.795.423
5.795.507
83
5,124,859
5,124,943
83
5,489,041
5,489,129
87
5,810,869
5.810,953
83
5,128,063
5.128,153
89
5.499.587
5,499,671
83
5.813.443
5.813.527
83
5.137.547
5.137.651
103
5.509,453
5.509.549 '
95
S.813.903
5,813,989
85
5.146,957
5,147,041
83
5.512,781
5.512,877
95
5.818,733
5,818,819
85
5,147,069
5.147.159
89
5.516,281
5.516,393
111
5.819.563
5,819,647
83
• 5.148,379
5,148,463
83
5.518,687
5.518,817
129
5,826,001
5.826,127
125
5,161,703
5,161,789
85
5.528,287
5.528,387
99
5.831.401
5.831,509
107
5.173.097
5.173.193
95
5.531.927
5.532,019
91
5.832,847
5,832,941
93
5.175.103
5.175,187
83
5.538,763
5.538,853
89
5,834,701
5.834,797
95
5.178,193
5,178,301
107
5.546,293
5.546,377
83
5.836,093
5.836,177
83
5.179.739
5.179.831
91
5.563.399
5.563.483
83
5,837.399
5.837.501
101
5.191.513
5.191.597
83
5,577.851
5.577.959
107
5.838,797
5.838,887
89
5,196,979
5.197.063
83
5,586,487
5.586,569
81
5.845.193
5.845.309
115
5.209,177
5,209,279
101
5,589,019
5.589.127
107
5,858,009
5.858,093
83
5,211,109
5,211,209
99
5,594,959
5.595.043
83
5,862,847
5,862,929
81
5,232,869
5.232,959
89
5,604,367
5,604,451
83
5.868,377
5,868,461
83
5,238,581
5.238,671
89
5,608,969
5,609,053
83
5,879,869 •
5.879.957
87
• 5.257,627
5.257.711
83
5.621,513
5.621,597
83
5.888,741
5,888,873
131
5,268,299
5.268,397
97
5.623,859
5,623,949
89
5,896,843
5,896,927
83
5,268,811
5,268,899
87
5.628,157
5,628,239
81
5,914,627
5.914.717
89
5.274,463
5.274.551
87
5.639.083
5.639.167
83
5.933.903
5,933.987
83
5.314,303
5.314.391
87
5,642,099
5,642,191
91
5.940,653
5.940,743
89
5.320,501
5.320,583
81-
5,646,131
5,646,229
97
5,942,777
5,942,863
85
5.321,507
5.321,593
85
5,649,221
5.649.307
85
5.950,963
5.951.051
87
5.323.949
5,324,051
101
5,651,623
5.651.707
83
5.967.347
5.967,433
85
5.332,759
5,332,841
81
5.656,837
5,656,921
83
5.976,079
5,976,181
101
5.335.189
5.335.271
81
5.671.279
5,671.361
81
5,982,413
5.982,523
109
5.337.217
5.337.307
89
5.673.449
5.673,557
107
5,989,999
5.990,081
81
5.339.903
5.339,987
83
5,695.531
5,695,619
87
5.341.463
5.341.559
95
5,723.899
5,723.999
99
The explanation of the table is that the 97 numbers between 5,001,809 and 5,001,907 are
composite, and so on — viz, the numbers in the columns headed " Lower limit " and " Upper
limit " are primes and the numbers intennediate to the lower limit and the upper limit are
composite.
§n.]
ENTTMEEATION OP PRIMES IN THE PIEST NINE MILLIONS.
63
The four O-prime centuries gave rise to sequences of 107, 129, 101, and 115 — ^viz. those
between 5,178,193 and 5,178,301, between 5,518,687 and 5,518,817, between 5,837,399 and
5,837,501, and between 5,845,193 and 5,845,309.
Of the double sequences only the four longest seem worth noting here: they are as
follows :— of the 165 numbers between 5,102,863 and 5,103,029 one only (5,102,953) is a prime ;
of the 149 numbers between 5,128,063 and 5,128,213 one only (5,128,153) is a prime ; of the
137 numbers between 5,438,831 and 5,438,969 one only (5,438,903) is a prime; and of the 133
numbers between 5,737,817 and 5,737,951 one only (5,737,889) is a prime.
The number of sequences of 81 and upwards given in the fourth million is 91, in the fifth
million is 89, and in the sixth million is 121.
The table was formed in the manner explained on p. 44 of the Introduction to the Fourth
Million. All the instances in which the enumeration showed that there were 0, 1 , 2, or 3 primes
in a century were looked out in the factor table, and the cases noted in which the difference
between two consecutive primes exceeded 49. Aflerwdxds the whole million was examined,
independently of the enumeration for sequences of 49 and upwards. Then, finally, from the long
list of sequences thus obtained, those of 81 and upwards were selected.
As in the case of the corresponding tables in the two previous millions the list is not given
as being necessarily complete, though I believe it to be so. The tables were formed independently
by different computers, and it seems almost impossible that any long sequences can have escaped
detection.
Seqnsnces in the Nine Millions.
The sequences of 81 and upwards for the fourth, fifth, and sixth millions were obtained in
the manner that has just been explained. With regard to the other three millions, a list of
sequences of 79 and upwards in the first million, and lists of sequences of 99 and upwards in the
second, third, seventh, eighth, and ninth millions were published in the Messenger of Mathematics*
for 1877 and 1878. These lists were obtained by a method similar to that employed in the case
of the other millions : they are not given as necessarily complete, although there can be no doubt
that they are very nearly so.
The table on the next page contains a list of the sequences exceeding 100 which occur in the
nine millions, those for the first three millions and the last three millions being taken from the
paper just referred to, those for the fourth and fifth millions from the tables on p. 45 and p. 10
of the Introductions to these Millions respectively, and those for the sixth million from the table
of sequences on the opposite page.
The table shows that the number of successions of more than a hundred composite numbers
occurring in the nine millions is 148, and that their distribution among the millions is as
follows : —
First million . .
. . 2
Fourth million .
. . 16
Second „
. . 11
Fifth
. . 16
Third „ . .
. . 10
Sixth „
. . 16
Seventh million
Eighth
Ninth
»>
27
23
27
" On Long Sncoeaaions of Composite Numbers," toI. vii. pp. 102-106 and 171-176.
64
INTEODTJCTION.
[§II,
List of Sequences exceeding 100 occubbing in the fibst nine millions.
Lower limit.
Upper limit
Sequence.
Lower limit
Upper limit
Sequence.
Lower limit
Upper limit
Sequenoe.
370*261
370,373
Ill
4,640*599
4,640,717
117
7,130,579
7,130,687
107
492,113
492,227
113
4,652,353
4,652,507
153
7,160,227
7,160,347
119
i.349»S33
1,349,651
117
4,686,709
4,686,811
101
7,227,551
7,227,667
115
1,357,201
1*357,333
131
4*738,651
4.738,777
125
7,230,331
7,230,479
147
1,388,483
1,388,587
103
4,958,021
4,958,131
109
7*293,989
7*294,093
103
1,444,309
1,444,411
101
5.137,547
5*137,651
103
7.309,427
7,309,529
101
1,468,277
1,468,387
109
5,178,193
5.178,301
107
7.345.967
7,346,069
101
1,561,919
1,562,051
131
5.209*177
5,209,279
101
7,445.047
7,445.159
111
1,664,123
1,664,227
103
5,323,949
5,324,051
101
7,494,763
7,494,869
105
1,671,781
1,671,907
125
5.516,281
5.516,393
111
7*565,191
7,565*303
111
1,761,187
1,761,289
101
5.518,687
5.518,817
129
7,621,259
7,621,399
139
i,775»o69
1,775*171
101
5.577.851
5.577.959
107
7,629,371
7,629,487
115
1*895,359
i,895>479
119
5,589,019
5,589.127
107
7,683,131
7,683,233
101
2,010,733
2,010,881
147
5.673.449
5.673.557
107
7,702,573
7,702,687
113
2,238,823
2,238,931
107
5,826,001
5,826,127
125
7.743,233
7,743.371
137
2,243,987
2,244,091
103
5,831,401
5.831.509
107
7.753.679
7.753.787
107
2,305,169
2,305,271
101
5.837.399
5.837.501
101
7.771.307
7.771*411
103
2.314,439
2,314*547
107
5.845.193
5.845.309
115
7.784.039
7.784*159
119
2,597,981
2,598,091
109
5*888,741
5,888,873
131
7.803,491
7,803,611
119
2,614,883
2,614,987
103
5,976,079
5*976,181
101
7,826,899
7,827,019
119
2,637,799
2,637,911
111
5,982,413
5*982,523
109
7,881,373
7.881,487
113
2,867,107
2,867,213
105
6,012,899
6,013,013
113
7,950,001
7,950,109
107
2,898,239
2,898,359
119
6,034,247
6*034*393
145
8,001,359
8,001,491
131
3,064,751
3,064,861
109
6*084,977
6,085,103
125
8,027,699
8,027,807
107
3,"7,299
3*"7,42i
121
6,085,441
6,085,561
119
8,073,647
8,073,749
101
3,225,539
3,225,647
107
6*158,563
6,158,681
117
8,166,107
8,166,209
101
3,240,983
3,241,093
109
6,333.799
6,333.907
107
8,172,713
8,172,821
107
3,279,841
3,279»949
107
6,347,059
6,347*177
117
8,208,449
8*208,553
103
3,359,"3
3.359*221
107
6,371,401
6,371,537
135
8,211,227
8*211,331
103
3*392,341
3.392,443
101
6,376,193
6*376,303
109
8,294,021
8,294,123
101
3,422,813
3,422,917
103
6,385.619
6,385,733
113
8,303,957
8,304,061
103
3,453,833
3.453,943
109
6,613,631
6,613,753
121
8,332,427
8*332*529
101
3,583,417
3,583,529
111
6,646,049
6,646,153
103
8,367,397
8,367*517
119
3,592,109
3.592,213
103
6,655,423
6,655,531
107
8,409,721
8409,829
107
3,593,203
3,593,3"
107
6,726,821
6,726,947
125
8,421,251
8,421,403
151
3,826,019
3,826,157
137
6,752,623
6,752,747
123
8,454.959
8,455.063
103
3,828,973
3,829,079
105
6,789.793
6,789,901
107
8,470,927
8,471*053
125
3,851,459
3.85i>587
127
6,808,273
6,808,397
123
8,487,421
8,487*527
105
3,933,599
3.933,731
131
6,812,233
6,812,339
105
8,514,949
8*515.063
113
4,044,077
4,044,179
101
6,826,159
6,826,271
111
8,648,557
8,648,677
119
4,131,109
4,131,223
118
6,836,867
6,836,969
101
8,752,871
8,752,987
115
4,166,893
4,166,999
105
6,845,417
6,845,519
101
8,793.881
8,793,991
109
4,234,537
4,234.651
113
6,877,109
6,877,219
109
8,850,671
8,850,773
101
4,297,093
4,297,199
105
6,879,683
6,879*797
113
8,889,031
8,889,143
111
4,315,607
4,315.709
101
6,958,667
6,958,801
133
8,905.199
8,905*321
121
4,447,321
4,447,423
101
6,972,127
6,972,233
105
8,917,523
8,917*663
139
4,478,423
4,478,527
103
6,983,843
6,983,947
103
8,939.597
8,939*699
101
4,535*717
4,535,819
101
6,991,147
6,991.253
105
8,947,217
8*947,319
101
4,536,179
4,536,283
103
7,129,877
7.130,003
125
8,981,461
8,981,563
101
4,596,731
4,596,833
101
It will be noticed that as early as 370,261 there is a sequence of 111, and that the longest
sequence met with in the nine millions is 153, which occurs about the middle of the fifth million.
In the sixth million there is no sequence which exceeds in length the two sequences of 131 which
occur in the second million.
§11.]
ENUMEEATION OF PEIMES IN THE FIBST NINE MILLIONS.
65
The following table contains the sequences exceeding 130 which occur in the nine millions,
arranged in the order of their length : —
to 9,000,000.
Sequences exceeding 130.
Lower limit.
Upper limit.
Sequence.
4,652,353
4,652,507
153
8,421,251
8,421,403
151
2,010,733
2,010,881
147
7,230,331
7,230,479
147
6,034,247
6,034,393
145
7,621,259
7,621,399
139
8,917,523
8,917,663
139
3,826,019
3,826,157
137
7,743,233
7,743,371
137
6,371,401
6,371,537
135
6,958,667
6,958,801
133
1,357,201
1,357,333
131
1,561,919
1,562,051
131
3,933,599
3,933,731
131
5,888,741
5,888,873
131
8,001,359
8,001,491
131
The two longest sequences occurring in each of the nine millions are as follows :-
Lower limit.
Tirst million | 37o,26i
I 492,113
f 1,357,20
1 1,561,91
{2,010,
2,898,
{3»
13,
Second
Third
Fourth
Fifth
Sixth
Seventh
Eighth
Ninth
201
9
733
239
826,019
933,599
{4,652,353
14,738,651
15,5^8,687 5i5i8,8i7
1 5,888,741 5,888,873
r 6,034,247
16,371,401
J 7,230,331
17,621,259
f 8,421,251
18,917,523
Upper limit.
Sequenoob
370,373
492,227
111
113
1,357,333
1,562,051
131
131
2,010,881
2,898,359
147
.... 119
3,826,157
137
3,933,731
131
4,652,507
4,738,777
153
125
5,518,817
5,888,873
129
131
6,034,393
6,371,537
146
135
7,230,479
7,621,399
147
139
8,421,403
8,9U,663
151
139
66
INTEODTJCTION.
[§m.
SECTION ni.
^COMPARISON OF THE RESULTS OF THE ENUMERATION WITH LEGENDRE'S,
TCHEBYCHEFFS, AND RIEMANN'S FORMULAE.
Legendre'a Formula.
The fifth section of the Introduction to the Fourth Million relates to the distribution of prime
numbers, and an account is there given of the different formulae which have been proposed for
representing approximately the number of primes inferior to a given number. The present
section contains the results of comparisons between the actual numbers of primes found by the
enumerations described in the preceding pages and the numbers given by these formulae.
The first formula for the approximate representation of the number of primes inferior to a
given prime x was given by Legendre in the second edition (1808) of his Essai sur la ThSorie
des Nombres (PeLTtlY. § viii. pp. 394-398). The Chapter is entitled "D'une loi tres-remarquable
observee dans I'enumeration des nombres premiers," and commences " Quoique la suite des
nombres premiers soit extremement irreguliere, on pent cependant trouver avec une precision
tres-satisfaisante, combien il y a de ces nombres depuis 1 jusqu'S. une limite donnee x. La
formule qui resout cette question est
^
y"-loga?-108366*
log X etant un logarithme hyperbolique."
Legendre then gives the following table, in which the values obtained from the formula are
compared with the numbers actually counted up to 400,000*: —
X,
Fonnulft.
Tables.
10,000
1230
1230
20,000
2268
2263
30,000
3252
3246
40,000
4205
4204
60,000
5136
5134
60,000
6049
6058
70,000
6949
6936
80,000
7838
7837
90,000
8717
8713
100,000
9588
9592
160,000
13844
13849
200,000
17982
17984
250,000
22035
22045
300,000
26023
25998
350,000
29961
29977
400,000
33854
33861
The enumerations were made from the factor table and list of primes in Vega's Tabulce,
which extend to 400,031.
After the table Legendre remarks : — " II est impossible qu'une formule represente plus
fiddlement une serie de nombres d'une aussi grande etendue et sujette necessairement k de
* Five of the values in the '' Tables " oolumn are in error to the extent of a unit, the correct numbers being 78^8,
8714, 9693, 17985, 29978.
§ III.]
LEGENDBE'S FOfiMtTLA.
67
frequentes anomalies ;" but he does not give any definite information with respect to the manner
in which he was led to the formula, or to assign the value 1'08366 to the constant. He does
indeed, at the end of the Chapter, indicate certain considerations from which he deduces that
the average interval between two primes at the point x in the series of numbers is of the form
Aloga;+B, and thence that the number of primes inferior to or is approximately equal to
X
Aloga?H-B— A'
"ce qui s'accorde avec la formule gen^rale donn^e ci-dessus, en prenant A=l, B= — 0-08366.*'
The reasoning, however, by which the average interval is found to be of the form A log^+B is
of a very vague and unsatisfactory character.
It would thus appear that Legendre, having been led by these analytical considerations, or in
some other way, to a formula of the form
A log a?— C'
determined the values of A and C empirically by means of the enumerations. The value of A
would at once be found to be unity; and since in 'the table reproduced above the number of
primes up to 10,000 is given as 1230 both from the formula and from the enumeration, and the
two numbers are identical for no other value of or, it would seem probable that the constant was
determined from the value ^=10,000. This, however, cannot have been the case, for if it were
so we should have
C=log, 1230-^^=1-08026,
which differs from 1'08366 by 0-00340. Legendre's formula for a:=10,000 gives 123051. Thus
although no doubt the constant was determined mainly from ^=10,000, it does not appear to
have been derived from any single value of x ; but it seems likely that it was so chosen as to
represent as nearly as possible the results of the entire enumerations.
In 1816 Legendre published a supplement to the ThSorie des NombreSj which contains
(pp. 61, 62) an addition to the Chapter on the distribution of primes. This consists principally
of the following continuation of the table from 400,000 to 1,000,000* :—
X.
Formula.
Tables.
400,000
33854
33863
600,000
41533
41538
600,000
49096
49093
700,000
56565
56535
800,000
63955
63937
900,000
71279
71268
1,000,000
78543
78493
The enumerations were made from Chemac's Cribrum Arithmeticuniy published in 1811,
which gives all the prime factors of numbers up to 1,020,000. It thus appears that when
Chemac's Cribrum afforded Legendre the means of extending his enumerations from 400,000 to
1,000,000, he availed himself of this opportunity of further testing his formula. The values
given by the formula agreed so well with the numbers counted over this extended range as to
apparently confirm the value which he had assigned to the constant, and he had therefore no
inducement to examine the question further or to redetermine the constant.
The correct values of the numbers in the "Tables" column are 33861, 41539, 49099, 66544, 63952, 71275, 78499.
k2
68
rNTEODTJCnON.
[§ III.
Taking the numbers as given by Legendre, the deviations are
X,
Deyiation.-
w.
DeriAtion.
10,000
O
200,000
— 2
20,000
5
250,000
— lO
30,000
6
300,000
25
40,000
I
350,000
-i6
50,000
2
400,000
- 7
60,000
- 9
500,000
- 5
70,000
13
600,000
3
80,000
I
700,000
30
90,000
4
800,000
i8
100,000
- 4
900,000
II
150,000
- 5
1,000,000
50
The smallness of these deviations is very striking and, considering the great irregularities in
the numbers of primes, it almost seems, as Legendre said, as if no formula could represent them
better ; and one can imagine how satisfactory the agreement must have appeared to him. It is
remarkable that a constant which might have been determined from ^=10,000 should so
accurately represent the numbers of primes up to 1,000,000.
In the third edition of the ThSorie des Nomhres (2 vols., 1830) the Chapter on the distribution
of primes remains substantially unaltered, but the two tables, viz. from to 400,000 and from
400,000 to 1,000,000, are united in a single table *.
In the second volume of Gauss's WerJce (1863) there is printed a letter from Gauss to Encke
dated December 24, 1849. An account of the contents of .this letter is given on pp. 30 and 31
of the Introduction to the Fourth Million. • Gauss mentions his own researches upon the
frequency of primes which date back to 1792 or 1793 ; and states that he soon recognized that in
spite of irregularities the average frequency was approximately inversely proportional to the
logarithm, so that the number of primes inferior to a given limit x was approximately
expressed by
Jdx
iog^'
the logarithm being hyperbolic. Having subsequently, with the assistance of Goldschmidt, made
an enumeration of the primes up to 3,000,000, Gauss compares the numbers for each half million
with the corresponding values given by the formula 1 ,r^— ; and he then refers to Legendre's
formula, which, he states, he had either overlooked or forgotten till Encke drew his attention
to it. Gauss then compares the numbers counted and the values given by Legendre's formula,
and finds the deviations to be
-23-3, +42-2, +68-1, +92-8, +159-1, +167-6;
and he remarks that although the deviations are less than in the case of the formula f,-^, they
Jlog^
seem to increase more rapidly. Writing Legendre's formula in the form
n
log«— a'
There \& a misprint in this table which does not occur in the second edition (1808), viz. the number of primes up to
300,000 is given as 25,988 instead of 26,998. In the table in the second edition (1808) the number of primes counted up
to 400,000 is given as 33,861, and in the supplement (1816) as 33,863 ; in the table in the third edition it is given as
33 861. In forming the deviations given in the text I have taken it as 33,861.
§ m,] TCHEBTCHEFF'S FOBMXILA. 69
he proceeds : — " Um Zahlung und Formel in Uebereinstimmung zu bringen, musste man respective
anstatt A=l-08366 setzen
1-09040, 107682, 1-07582, 1-07529, 1-07179, 1-07297.
" Es scheint, dass bei wachsendem n der (Durschnitts-) Werth von A abnimmt, ob aber die
Grenze beim Wachsen des n ins Unendliche 1 oder eine von 1 verschiedene Grosse sein wird,
dariiber wage ich keine Vermuthung."
Gauss thus determines the value of the constant by putting ar=500,000,ar=l,000,000, . . .
ar= 3,000,000 in the same manner as we may suppose that Legendre determined it originally by
putting ar=10,000.
The enumerations employed by Gauss are, as has been pointed out on p. 55, very inaccurate ;
and if the correct values of the number of primes be taken, the deviations are found to be
-6-3, +44-2, +26-6, +43-8, +102-1, +95-5,
instead of those given by Guuss. The corresponding values of A are
1-08548, 1-07649, 1-08061, 1-07971, 107605, 107757.
The correct values of A do not therefore diminish so rapidly as those given by Gauss. The
largest value of A is 1085 and the least 1-076, the range being 0009 ; but Gauss's largest value
is 1-090 and his least 1-072, the range being 0-018*.
Tchebycheff's Formula.
In 1848 Professor Tchebycheflf communicated to the Academy of St. Petersburg a memoir,
** Sur la fonction qui determine la totalite des nombres premiers inferieurs k une limite donnee,"
which is referred to on p. 32 of the Introduction to the Fourth Million. He there investigates
limits between which f{x\ the total number of primes inferior to ^, must lie, and shows that
when w is very large we may put approximately
A^)-i
2 l^g ^
In the following year the late Mr. C. J. Hargreave published a paper in the Philosophical
Magazine in which he represented the number of primes between x and a! by the formula
C' dx
J^ log X
The integral I j-— is termed the logarithm-integral, and is written li x. It thus appears
that the function li x was employed by Gauss, Tchebycheflf, and Hargreave independently to
represent approximately the number of primes inferior to x. The first discovery is no doubt due
to Guuss, but Tchebycheflf was the first to publish the connexion between li x and the numbers
of primes.
The logarithm-integral is defined by the equation
dx
log x^
and therefore contains an infinite element corresponding to a?=l which renders the value of the
integral infinite ; this element is, however, supposed to be omitted, and the principal value taken,
llie actual formula given by Tchebycheflf is li ^— li 2, in which the infinite element is not included
* Assuming Gauss's values for the number of primes counted, his values of A are correctly deduced except in the case
of 1,500,000, for which the value of A should be 1*07600 instead of 1*07582. It seems not improbable tha£ Gauss's value
was 1*07599, and that in writing to Encke he inadvertently copied 82, the last two figures of the previous number, instead
of 99.
^^^=r
70
INTEODUCTION.
[§UI.
between the limits of integration. The value of li 2 is 1*04516 . . . . , so that if (p{x) denote the
number of primes inferior to x including unity as a prime, and f{x) denote the number when unity
is excluded, ]ix^(f>{s) and (liar— li 2)—f(x) diflfer from one another only by 0045 . • . In all the
tables which follow, unity is counted as a prime and the comparison is made with li x.
Comparison of the Numbers of Primes with Legendre's and Tchehycheff's Formulas.
Table I. (pp. 71-73) shows the numbers of primes counted for values of x differing by
50,000 up to 9,000,000, and also the corresponding values given by Legendre's formula and by the
formula li x. The deviations are shown in the next two columns, and the last column contains
the differences between the values given by the two forniute.
Table II. (pp. 74-76) shows the numbers of primes in each group of 50,000, and also the
values given by each of the two formula? for these groups. The arrangement is exactly the same
as in Table I. ; and indeed the results in Table II. are the differences of the results in Table I.,
or, conversely. Table I. may be regarded as formed by continued addition from Table II.
It may be remarked here that the numbers calculated by the formulse and also the differences
are given correctly to the nearest unit, so that there are sometimes discrepancies of a unit between
the differences given and the numbers obtained by subtraction from the columns in the tables.
In all cases the values derived from the formulae were calculated to at least one decimal place,
and generally to two or three, and the numbers given in the tables are those obtained by
contraction from these values.
It will be seen from Table I. that up to about 5,000,000 Legendre's formula gives the better
result, but that for higher values of x the deviations are less in the case of the formula liar.
Legendre's formula contains a disposable constant, which is so determined as to make the formula
agree with the numbers counted when x is small ; and we therefore should expect that at first
Legendre's formula would give more satisfactory results than li x. In the latter part of the table
it gives a result which is considerably too large, and it is evident that, on the whole, li x is much
the more satisfactory formula of the two.
When ar=10, w^ZTo8366~^*^^' ^^^ liar=6-17 ; for ar=50,000, the values given by the
two formulsB differ by 31 ; the difference then increases to about 85, from ar= 1,050,000 to
ar=l,450,000 ; it then decreases to zero at a:= 4,850,000; and after this the formulae diverge
continually, and differ by 174 at ar= 9,000,000. It is thus evident that comparisons between
the formulae, unless they extend beyond 5,000,000, might be very misleading *.
The last column of Table II. shows how close is the agreement between the values given by
the two formulae for ranges of 50,000. Beyond 400,000 the difference never exceeds 2. At
first the formula li x gives the greater value ; between 1,000,000 and 2,400,000 the values are
identical ; and for higher numbers the greater value is given by Legendre's formula.
.^
* Tchebycheff remarks in the memoir already cited that
is a minimum when
X C' dx
log a?— 1-08366 J, logi
(1'08866)«
a.=:« 008366 « 1,247,646,
but increases rapidly above this limit, and for jp= 10,000,000 is considerable, so that for this value of x the advantage of one
formula over the other ought to be apparent. The difference, however, is a maximum, not a minimum, when jr= 1,247,646 ;
and beyond this point the formulsB approach one another, and give identical results when a? s= 4,850,000 nearly ; it is only
then that the numerical difference between them begins to increase.
§UI.]
COMPABISON WITH LEGENDRE'S AND TOHEBTCHEFP'S POEMTTliaE.
71
TABLE I. — COMPAEISON with LeGENDRE'S and TcHEBYCHEFP's FOBMULfli AT INTERVALS OF
50,000 DP TO 9,000,000.
Number of Primes
Difference
^H V •
Between numbers ooimted and
Jr.
OoimtecL
Caloulated by
caloulated by
Between the
X
]ix.
X
two formuln.
logx-1-08366'
loga?--l-08366*
u » •
50,000
5134
5136
5167
+ 2
+ 33
-31
100,000
9593
9588
9630
- 5
37
41
150,000
13849
13844
13893
- 5
44
49
200,000
1798s
17982
18036
- 3
51
53
250,000
22045
22035
22094
— 10
49
59
300,000
25998
26024
26087
+ 26
89
63
350,000
29978
29961
30027
- 17
49
66
400,000
33861
33854
33923
- 7
62
69
450,000
37707
37709
37781
+ 2
74
71
500,000
41539
41533
41606
- 6
67
73
550,000
45323
45327
45403
4- 4
80
75
600,000
49099
49096
49173
- 3
74
77
650,000
52832
52841
52920
+ 9
88
79
700,000
56544
56565
56645
+ 21
lOI
80
750,000
60239
60269
60350
+ 30
III
81
800,000
63952
63955
64037
+ 3
85
82
850,000
67618
67625
67708
+ 7
90
83
900,000
7127s
71279
71362
+ 4
87
83
950,000
74908
74918
75002
+ 10
94
84
1,000,000
78499
78543
78628
+ 44
129
84
1,050,000
82135
82156
82240
+ 21
105
85
1,100,000
85715
85756
85841
+ 41
126
85
1,150,000
89304
89344
89429
4. 40
125
85
1,200,000
92940
92921
93007
- 19
67
85
1,250,000
96470
96488
96573
+ 18
103
85
1 1,300,000
I 0002 I
100045
IOO130
24
109
85
1 1,350,000
103543
103592
103677
49
134
85
1 1,400,000
107124
107129
107214
5
90
85
1 1,450,000
110626
I 10658
I 10743
32
117
85
[ 1,500,000
114152
114179
I 14263
27
III
84
i.55o>ooo
117660
I17691
"7775
31
. "5
84
3,600,000
121125
121195
121279
70
154
84
-1,650,000
124633
124692
124775
59
142
83
:*, 700,000
128140
128181
128264
41
124
83
::^, 750,000
131607
131663
131746
56
139
82
j^r,3oo,ooo
135072
135139
135221
67
149
82
;^C' ,850,000
138542
138608
138689
66
147
81
^T ,^00,000
142029
142070
142150
41
121
80
er ^^505000
145502
145526
145606
24
104
80
r ^^«:> 00,000
148932
148976
149055
44
123
79
r ^^<=:> 50,000
152382
152420
152498
38
116
78
-^ :k: 00,000
155806
155858
155936
52
130
77
-^ ^^ 50,000
159251
159291
159368
40
117
77
•9 ^^ 00,000
162663
162718
162794
55
131
76
» ^^ ^0,000
166082
166140
166215
58
133
75
-^ .^S- 00,000
169512
169557
I 6963 I
45
119
74
•»^C^ 50,000
172874
172969
173042
95
168
73
^^'^^ 00,000
176303
176376
176448
73
145
72
'»-'==• 50,000
179685
179778
179849
93
164
71
-» -S 00,000
183073
183175
183245
102
172
70
».S 50,000
186464
186568
186637
104
173
69
»^^ 00,000
189882
189956
190024
74
142
68
*^^ 50,000
193258
193340
193407
82
149
67
[ * ^ oo,ooo
196647
196720
196785
73
138
65
\^^. -50,000
•'^^^0,000
*^ 30,000
199995
200095
200160
100
165
64
203363
203467
203530
104
167
63
206790
206834
206896
44
106
62
* ^ ^=^Ci,000
210109
210197
210258
88
149
61
*^3ci,ooo
213454
213557
213616
103
162
59
*^^^^^c:>,ooo
216817
216913
216971
+ 96
+ 154
-58
72
INTEODirCTION.
[§ni.
TABLE I. {continued).
X,
Number of Primes
Difference
Counted.
Caloulated by
Between numbers counted and
calculated by
Setwf^m the
X
Mx.
X
U*.
two formula.
logx-108366'
logjr-1-08366*
3,050,000
220137
220264
220321
+ 127
+ 184
-57
3,100,000
223493
. 223613
223668
120
175
55
3,150,000
226836
226958
227012
122
176
54
3,200,000
230210
230299
230351
89
141
53
3,250,000
233578
233636
233688
58
no
52
3,300,000
236901
236971
237021
70
120
50
3»350iOoo
240231
240302
240350
71
119
49
3,400,000
243540
243629
243677
89
137
47
3,450,000
246910
246954
247000
44
90
46
3,500,000
250151
250275
250319
124
168
44
3,550,000
253412
253593
253636
181
224
43
3,600,000
256726
256908
256950
182
224
41
3,650,000
260064
260220
260260
156
196
40
3,700,000
263397
263529
263568
132
171
38
3,750,000
266717
266835
266872
118
155
37
3,800,000
269987
270139
270174
152
186
35
3,850,000
273322
273439
273473
117
151
34
3,900,000
276611
276737
276769
126
158
32
3,950,000
279921
280031
280062
no
141
31
4,000,000
283146
283323
283352
177
206
29
4,050,000
286490
286613
286640
123
150
27
4,100,000
289774
289899
289925
125
151
26
4,150,000
293010
293184
293208
174
198
24
4,200,000
296314
296465
296487
151
173
22
4,250,000
299583
299744
299765
161
182
21
4,300,000
302824
303020
303039
196
215
19
4»35o»ooo
306084
306294
306312
210
228
17
4,400,000
309335
309566
309582
231
247
16
4,450,000
312666
312835
312849
169
183
14
4,500,000
315948
316102
316114
154
166
12
4i55o,ooo
319164
319366
319377
202
213
10
4,600,000
322441
322628
322637
187
196
9
4,650,000
325706
325888
325895
182
189
7
4,700,000
328964
329145
329150
181
186
5
4,750,000
332219
332400
332404
181
185
4
4,800,000
335439
335653
335655
214
216
— 2
4,850,000
338695
338904
338904
209
209
4,900,000
341993
342153
342151
160
158
+ 2
4,950,000
345236
345399
345396
163
160
4
5,000,000
348515
348644
348638
129
123
6
5,050,000
351717
351886
351879
169
162
7
5,100,000
354973
355126
355117
153
144
9
5,150,000
358200
358364
358353
164
153
II
5,200,000
361409
361601
361588
192
179
13
5,250,000
364687
364835
364820
148
^ZZ
15
5,300,000
367902
368067
368050
165
148
17
5>35o»ooo
371133
371297
371279
164
146
»9
5,400,000
374364
374525
374505
161
141
21
5,450,000
377576
377752
377729
176
153
22
5,500,000
380802
380976
380952
174
150
24
5,550,000
383981
384199
384173
218
192
26
5,600,000
387204
387419
387391
215
187
28
5,650,000
390380
390638
390608
258
228
30
5,700,000
393608
393855
393823
247
215
32
5,750,000
396828
397071
397037
243
209
34
5,800,000
399995
400284
400248
289
253
3(^
5,850,000
403206
403496
403458
290
252
38
5,900,000
406431
406706
406666
275
235
40
5,950,000
409674
409914
409872
240
198
42
6,000,000
412851
413121
413077
4-270
+ 226
+44
§ III.]
COMPAEISON WITH LEGENDRE'S AND TCHEBTCHEFF'S FOSMULS.
73
TABLE I. (continued).
4e.
6,050,000
6,100,000
6,150,000
6,200,000
6,250,000
6,300,000
6,350,000
6,400,000
6»45090oo
6,500,000
6,sso,ooo
6,600,000
6,650,000
6,700,000
6,750,000
6,800,000
6,850,000
6,900,000
6,950,000
7,000,000
7,050,000
7,100,000
7,150,000
7,200,000
7,250,000
7,300,000
7»35o>ooo
7,400,000
7,450,000
7,500,000
7i5So>ooo
7,600,000
7,650,000
7,700,000
7,750,000
7,800,000
7,850,000
7,900,000
7,950,000
8,000,000
8,050,000
8,100,000
8,150,000
8,200,000
8,250,000
8,300,000
8,35o»ooo
8,400,000
8,450,000
8,500,000
8,550,000
8,600,000
8,650,000
8,700,000
8,750,000
8,800,000
8,850,000
8,900,000
8,950,000
9,000,000
Number of Frimet
Counted
416042
419248
422451
425650
428818
432075
435267
438412
441564
444759
447957
451161
454339
457499
460709
463874
467057
470285
473472
476650
479868
483019
486173
489325
492501
495673
498805
501972
505158
508273
5^429
514578
517754
520925
524041
527170
530354
533534
536683
539808
542929
546058
549185
552359
555520
558642
561813
564927
568019
571172
574332
577498
580627
583779
586918
590078
593186
596298
599433
602568
Oaloulated by
]og«-l-08d66'
lix.
416325
419528
422730
425930
429128
432324
435519
438712
441904
445094
448283
451470
454655
457839
461022
464203
467382
470560
473737
476912
480086
483258
486429
489598
492766
495933
499099
502263
505425
508587
511747
5^905
518062
521219
524373
527527
530679
533830
536980
540128
543276
546422
549567
552710
555853
558994
562134
565273
568410
571547
574682
577817
580950
584082
587213
590342
593471
596599
599725
602850
416279
419480
422680
425878
429074
432268
435461
438652
441842
445030
448216
451401
454585
457767
460947
464126
467303
470479
473654
476827
479998
483169
486337
489505
492671
495835
498998
502160
505321
508480
511638
514794
517949
521103
524256
527407
530557
533706
536853
540000
543145
546289
549431
552573
555713
558852
561990
565126
568262
571396
574529
577661
580792
583921
587050
590178
593304
596429
599553
602676
Differenee
1
Between numbers counted and
calculated by
log*-l-08366'
Vlx,
+ 283
280
279
280
310
249
252
300
340
335
326
309
316
340
313
329
325
275
265
262
218
239
256
273
265
260
294
291
267
314
318
327
308
294
332
357
325
296
297
320
347
364
382
351
333
352
321
346
391
375
350
319
323
303
295
264
285
301
292
+282
+ 237
232
229
228
256
193
194
240
278
271
259
240
246
268
238
252
246
194
182
177
130
150
164
180
170
162
193
188
163
207
209
216
195
178
215
237
203
172
170
192
216
231
246
214
193
210
177
199
243
224
197
163
165
142
132
100
118
131
120
+ 108
Between the
two formnliB.
+ 46
48
50
52
54
56
58
60
62
64
66
68
70
72
75
77
79
81
83
85
87
89
92
94
96
98
100
102
105
107
109
III
"3
"5
118
120
122
124
127
129
131
^33
^3^
138
140
142
144
147
149
152
154
156
158
161
163
165
167
170
172
+ 174
74
INTBODUCnON.
[§ra,
TABLE II. — CoMPABisoK with Legbndre's and Tchebychbpp's FoEMULiE FOE Geoups op
50,000 UP TO 9,000,000.
Number of Primes
jyiffemnoo
^^ m V
Between numbers counted and
Limits.
Calculated by
calculated by
Counted.
w
Between the
two formula.
X
lix
X
Mx
log X- 108366'
At •••
logar-1-08366'
mX mrm
o- 50,000
5134
5136
5167
+ 2
+33
-31
50,000- 100,000
4459
4453
4465
- 6
+ 6
12
100,000- 150,000
4256
4256
4264
+ 8
8
i5°>ooo~ 200,000
4136
4137
4143
+ I
+ 7
6
200,000- 250,000
4060
4053
4058
- 7
- 2
5
2 5 Oi 000- 3 00,000
3953
3989
3993
+ 36
+40
4
300,000- 350,000
3980
3937
3940
-43
—40
3
350,000- 400,000
3883
3893
3896
+ 10
+13
3
400,000- 450,000
3846
3856
3858
+ 10
+ 12
2
45o»ooo- 500,000
3832
3823
3825
- 9
- 7
2
500,000- 550,000
3784
3795
3796 .
■fii
+ 12
2
550,000- 600,000
3776
3769
3770
- 7
- 6
2
600,000- 650,000
3733
3745
3747
+ 12
+14
2
650,000- 700,000
3712
3724
3725
+ 12
+13
700,000- 750,000
3695
.3704
3705
+ 9
+ 10
750,000- 800,000
3713
3686
3687
-27
-26
800,000- 850,000
3666
3669
3670
+ 3
+ 4
850,000- 900,000
3657
3654
3654
- 3
- 3
900,000- 950,000
3633
3639
3640
+ 6
+ 7
950,000- 1 ,000,000
3591
3625
3626
+ 34
+35
— I
1,000,000-1,050,000
3636
3612
3613
— 24
-23
1 ,050,000-1 , 1 00,000
3580
3600
3600
4-20
+ 20
1,100,000-1,150,000
3589
3588
3589
— I
1,150,000-1,200,000
3636
3577
3577
-59
-59
1,200,000-1,250,000
3530
3567
3567
+37
+37
1,250,000-1,300,000
3551
3556
3557
+ 5
+ 6
1,300,000-1,350,000
3522
3547
3547
+ 25
+ 25
I >3 50*000- 1, 400,000
3581
3538
3538
-43
-43
1 ,400,000-1 ,450,000
3502
3529
3529
+ 27
+ 27
1 ,450,000- 1 ,500,000
3526
3520
3520
- 6
- 6
1,500,000-1,550,000
3508
3512
3512
+ 4
+ 4
I »5 50*000-1 ,600,000
3465
3504
3504
+ 39
+39
1,600,000-1,650,000
3508
3497
3496
— II
— 12
1,650,000-1,700,000
3507
3489
3489
-18
-18
1,700,000-1,750,000
3467
3482
3482
+ 15
+ 15
I >7 50,000- 1, 800,000
3465
3475
3475
+ 10
+ 10
1,800,000-1,850,000
3470
3469
3468
— I
- 2
1,850,000-1,900,000
3487
3462
3462
-25
-25
1,900,000-1,950,000
3473
3456
3455
-17
-18
1 ,950,000-2,000,000
3430
3450
3449
+ 20
+ 19
2,000,000-2,050,000
3450
3444
3443
- 6
- 7
2,050,000-2,100,000
3424
3438
3438
+ 14
+ 14
2,100,000-2,150,000
3445
3433
3432
— 12
-13
2,150,000-2,200,000
3412
3427
3426
+ 15
+ 14
2,200,000-2,250,000
3419
3422
3421
+ 3
+ 2
2,250,000-2,300,000
3430
3417
3416
-13
-14
2,300,000-2,350,000
3362
3412
341 1
+ 50
+49
2,350,000-2,400,000
3429
3407
3407
— 22
— 22
2,400,000-2,450,000
3382
3402
3401
4-20
+ 19
+ I
2,450,000-2,500,000
3388
3397
3396
+ 9
+ 8
2,500,000-2,550,000
3391
3393
3391
+ 2
2,550,000-2,600,000
3418
3388
3387
-30
-31
2,600,000-2,650,000
3376
3384
3383
+ 8
+ 6
2,650,000-2,700,000
3389
3380
3379
- 9
— 10
2,700,000-2,750,000
3348
3375
3374
+ 27
+ 26
2,750,000-2,800,000
3368
3371
3370
+ 3
+ 2
2,800,000-2,850,000
3427
3367
3366
-60
-61
2,850,000-2,900,000
3319
33^3
3362
+44
+43
2,900,000-2,950,000
3345
3359
3358
+ M
+ 13
2,950,000-3,000,000
33^3
3356
3354
- 7
- 9
+ I
$ra.]
COMPAEISON WITH LBGENDEB'S AND TCHBBTOHEEP'S FOEMXJLE.
75
TABLE IL {continued).
^ • •«
Number of Primes.
DiiEarenoe
•
^^ • V
Between numbers oounted and
liinuto.
Caloulfttea bj {
calculated I7
Oounted.
•
Between the
two formuln.
s
\xx.
X
\\x
logx- 1*06366'
logor-i-osaee'
U Wc
3,000,000-3,050,000
3320
3352
3351
+32
+31
+1
3,050,000-3,100,000
3356
3348
3347
- 8
— 9
3,100,000-3,150,000
3343
3345
3343
+ 2
3,150,000-3,200,000
3374
3341
3340
-33
-34
3,200,000-3,2 50,000
3368
3338
ZZ3^
—30
-32
3,2 50,000-3,300,000
3323
3334
3333
+ 11
+ 10
3»3oo»ooo-3,3 50,000
. 3330
3331
3330
+ I
3»35o»ooo-3*4oo>ooo
3309
3328
3326
+ 19'
+ 17
3,400,000-3,450,000
3370
3325
3323
-45
-47
3945o»ooo-3»5oo,ooo
3241
3321
3320
+80
+ 79
3*5001000-3,550,000
3261
3318
3317
+ 57
+ 56
3>55o>ooo-3,6oo,ooo
3314
3315
3314
+ I
3,600,000-3,650,000
3338
3312
33"
-26
-27
3*650,000-3,700,000
3333
3309
3308
-24
-25
3,700,000-3,750,000
3320
3306
3305
-14
-15
3»75o>ooo-3,8oo,ooo
3270
3303
3302
+ 33
+32
3,800,000-3,850,000
3335
3300
3299
-34
-36
3*850,000-3,900,000
3289
3297
3296
+ 8
+ 7
3,900,000-3,950,000
3310
3295
3293
-15
-17
2
3>95o>ooo-4,ooo,ooo
3225
3292
3290
+67
+65
2
4,000,000-4,050,000
3344
3289
3288
-55
-56
2
4*050,000-4* 1 00,000
3284
3287
3285
+ 3
+ I
2
4,100,000-4,1 50,000
3236
3284
3282
+ 48
+46
2
4,1 50,000-4,200,000
3304
3281
3280
-23
-24
2
4,200,000-4,250,000
3269
3279
3277
+ 10
+ 8
2
4,250,000-4,300,000
3241
3277
3275
+36
+34
2
4,300,000-4,350,000
3260
3274
3272
+ 14
+ 12
2
4,350,000-4,400,000
3251
3272
3270
+ 21
+ 19
2
4,400,000-4,450,000
3331
3269
3267
-62
-64
2
4,450,000-4,500,000
3282
3267
3265
-15
-17
2
4,500,000-4,5 50,000
3216
3265
3263
+49
+ 47
2
4,550,000-4,600,000
3277
3262
3260
-15
-17
2
4,600,000-4,650,000
3265
3260
3258
- 5
- 7
2
4,650,000-4,700,000
3258
3257
3256
— I
— 2
2
4*700,000-4,750,000
3255
3255
3253
— 2
•
2
4,7 50,000-4,800,000
3220
3253
3251
+33
+31
2
4,800,000-4,850,000
3256
3251
3249
- 5
- 7
2
4,850,000-4,900,000
3298
3249
3247
-49
-SI
2
4,900,000-4,950,000
3243
3246
3245
+ 3
+ 2
2
4,950,000-5,000,000
3279
3244
3243
-35
-36
2
5,000,000-5,050,000
3202
3242
3240
+40
+38
2
5,050,000-5, 100,000
3256
3240
3238
-^16
-18
2
5,100,000-5,150,000
3227
3238
3236
+ 11
+ 9
2
5,150,000-5,200,000
3209
3236
3234
+ 27.
+ »5
2
5,200,000-5,250,000
3278
3234
3232
-44
-46
2
5,250,000-5,300,000
3215
3232
3230
+ 17
+ 15
2
5,300,000-5,350,000
3231
3230
3228
— I
- 3
2
5*3 50,000-5 ,400,000
3231
3228
3226
- 3
- 5
2
5^400,000-5,450,000
3212
3226
3224
+ 14
+ 12
2
5*450,000-5,500,000
3226
3225
3223
— I
- 3
2
5*500,000-5,550,000
3179
3223
3221
+ 44
+4*
2
5*550,000-5,600,000 .
3223
3221
3219
— 2
— 4
2
5,600,000-5,650,000
3176
3219
3217
+ 43
+41
2
5,650,000-5,700,000
3228
3217
3215
— II
-13
2
5,700,000-5,750,000
3220
3215
3213
- 5
- 7
2
5*750,000-5,800,000
3167
3214
3211
+47
+44
2
5,800,000-5,850,000
3211
3212
3210
+ I
— 1
2
5,850,000-5,900,000
3225
3210
3208
-IS
-17
2
5,900,000-5,950,000
3243
3208
3206
-35
-37
2
5,950,000-6,000,000
3177
3207
3204
+ 30
+ 27
+ 2
l2
76
INTRODUCTION.
[§ m.
TABTiE II
. {continued).
Number of Primes.
•
Differenoe
..
Between numbers oounted and
limits.
Oounted.
Calculated by
calculated by
Between the
two formula.
T
Vi».
X
Uf.
log X- 1-08366'
log«-l-08366'
6,000,000-6,050,000
3191
3205
3203
+ 14
+ 12
+ 2
6,050,000-6, 100,000
3206
3203
3201
- 3
- s
2
6, 1 00,000-6, 1 50,000
3^03
3202
3199
— I
- 4
2
6, 1 50,000-6,200,000
3199
3200
3198
4- I
— I
2
6,200,000-6,250,000
3168
3198
3196
4-30
4-28
2
6,2 50,000-6,300,000
3257
3197
3194
-60
-63
2
6,300,000-6,3 50,000
3192
3'95
3193
+ 3
+ I
2
6,3 50,000-6,400,000
3H5
3^93
3191
-1-48
+46
2
6,400,000-6,450,000
3152
3192
3190
+40
+38
2
6,450,000-6,500,000
3195
3190
3188
- S
- 7
2
6,500,000-6,550,000
3198
3188
3187
— 10
— II
2
6>55o>ooO""6>^oo,ooo
3204
3187
3185
-17
-19
2
6,600,000-6,650,000
3178
3185
3183
+ 7
4 5
2
6,650,000-6,700,000
3160
3184
3182
+ 24
+ 22
2
6,700,000-6,750,000
3210
3183
3180
-27
-30
2
6,7 50,000-6,800,000
3165
3181
3179
+ 16
+ 14
2
6,800,000-6,850,000
3^^3
3179
3177
— 4
— 6
2
6,8 50,000-6,900,000
3228
3178
3176
-50
-52
2
6,900,000-6, 9 5 0,000
3187
3177
3174
— 10
-13
2
6,950,000-7,000,000
3178
3175
3173
- 3
- 5
2
7 ,000,000-7 ,0 50,000
3218
3174
3172
-44
-46
2
7 »o5o>ooo-7, 1 00,000
3151
3172
3170
+ 21
4-19
2
7,100,000-7,150,000
3154
3171
3169
+ 18
4-15
2
7,150,000-7,200,000
3152
3170
3167
4-18
+ 15
2
7,200,000-7,250,000
3176
3168
3166
- 8
— 10
2
7,250,000-7,300,000
3172
3167
3165
- S
- 7
2
7,300,000-7,350,000
3132
3165
3^^3
•^33
+ 31
2
7 93 50^000-7 ,400,000
3167
3164
3162
- 3
- 5
2
7,400,000-7,450,000
31^6
3^^3
3160
-23
-26
2
7*450,000-7,500,000
3"5
3161
3159
+46
+44
2
7,500,000-7,550,000
3156
3160
3158
+ 4
+ 2
2
7»55o»ooo-7,6oo,ooo
3H9
3159
3156
+ 10
+ 7
2
7 ,600,000-7 »6 5 jOoo
3176
3157
3155
-19
—21
2
7,650,000-7,700,000
3171
315^5
3154
-15
-17
2
7 , 7 00,000- 7,75 0,000
3116
315s
3153
+39
+ 37
2
7i75o»ooo-7,8oo,ooo
3129
3153
3151
+ 24
422
2
7 ,800,000-7 ,8 50,000
3184
3152
3150
-32
-34
2
7 ,8 50,000-7 ,900,000
3180
3151
3M9
-29
-31
2
7,900,000-7,950,000
3149
3150
3148
+ I
— I
2
7>95o»ooo-8,ooo,ooo
3125
3149
3146
4-24
+ 21
2
8,000,000-8,050,000
3121
3147
3145
+ 26
+ 24
2
8,050,000-8, 1 00,000
3129
3146
3144
+ 17
+ 15
2
8, 100,000-8, 1 50,000
3127
3145
3143
+ 18
+ 16
2
8, 1 50,000-8,200,000
3'74
3144
3141
-30
-33
2
8,200,000-8,250,000
3161
3142
3140
-19
—21
2
8,2 50,000-8,300,000
3122
3141
3^39
-I-I9
+ 17
2
8,300,000-8,350,000
3171
3140
3138
-31
-33
2
8,3 50,000-8,400,000
3114
3139
3137
+25
+ 23
2
8,400,000-8,450,000
3092
3137
3135
4-45
+ 43
2
8,450,000-8,500,000
3153
3136
3134
-'7
-19
2.
8,500,000-8,550,000
3160
3135
3^3
-25
-27
2
8,550,000-8,600,000
3166
3134
3132
-32
-34
2
8,600,000-8,650,000
3129
3^33
3131
+ 4
+ 2
2
8,650,000-8,700,000
3152
3132
3130
— 20
—22
2
8,700,000-8,750,000
3139
3^3^
3129
— 8
— 10
2
8,750,000-8,800,000
3160
3^30
3127
-30
+ 21
-33
2
8,800,000-8,850,000
3108
3129
3126
+ 18
2
8,850,000-8,900,060
3112
3128
3125
+ 16
+ 13
2
8,900,000-8,950,000
3135
3126
3124
- 9
— II
2
8,950,000-9,000,000
3135
3125
3123
— 10
— 12
+ 2
I
§ni.]
CALCULATION OF THE CONSTANT IN LEGENDEE'S FOEMULA.
77
X
In order to exhibit the yariation in A, the constant in the formula . «„a > ^cording to
the value of x by means of which A is determined, the following table wcui calculated. It gives,
in the first column of results, the values of A as determined from ar= 50,000, a?= 100,000, . . .
07=9,000,000. The next two columns contain the differences between the values of A and
Legendre's value 1*08366, and the differences between the numbers of primes counted and the
values given by Legendre's formula. The last column gives the differences in the value of A
caused by a change of 1 in the number of primes counted. Thus, for example, a change of 1 in
the number of primes in the first 50,000 numbers would have changed the constant, if derived
from a:=50,000, by 0*00190, and a change of 1 in the number of primes in the first 9,000,000
numbers would have changed it by 000002, if derived from ar= 9,000,000. The numbers in this
column, except at the beginning of the table, are nearly equal to the quotients of the numbers
in the two preceding columns.
TABLE III. — Calculation of A at intervals op 50,000 up to 9,000,000.
Difference
Difference
Difference
Difference
between
in A pro-
duced by
between
in A pro-
duced by
number of
number of
1-08366
primes and
difierenoe
1-08366
primes and
difference
X.
A
-A.
number
given by
Legendre's
of 1 in
number of
primes
X,
A.
-A.
number
given by
TiOgendre's
of 1 in
number of
primes
formula.
counted.
formula.
counted.
50,000
i^o8o78
-f 288
+ 2
190
2,150,000
1-08023
+ 343
+ 40
8
100,000
ro8866
— 500
- s
109
2,200,000
1-07907
459
55
8
1 50,000
1-08728
— 362
- 5
78
2,250,000
1-07892
474
58
8
200,000
1-08576
— 210
- 3
62
2,300,000
1-08005
361
45
8
250,000
1-08878
- 512
—10
51
2,35o>ooo
1-07621
745
95
8
300,000
i"072i9
+ 1147
+26
44
2,400,000
1-07805
561
73
8
350,000
i^09046
- 680
-17
39
2,450,000
1-07663
703
93
8
400,000
1-08622
— 256
- 7
35
2,500,000
1-07605
762
102
450,000
1-08288
+ 78
+ 2
32
2,550*000
1-07604
762
104
500,000
1-08548
- 182
- 6
29
2,600,000
1*07831
535
74
550,000
1-08255
4- III
+ 4
27
2,650,000
1-07783
583
82
600,000
1-08448
- 82
- 3
25
2,700,000
1-07858
508
73
650,000
1-08158
4- 208
+ 9
23
2,750,000
1-07677
689
100
700,000
1-07910
+ 456
+21
22
2,800,000
1-07670
696
104
750,000
1-07742
4- 624
+ 30
21
2,850,000
1-08073
293
44
800,000
1-08299
+ 67
+ 3
20
2,900,000
1-07786
580
88
850,000
1-08237
+ 129
+ 7
19
2,950,000
1-07701
665
103
6
900,000
1-08300
-1- (^(^
+ 4
18
3,000,000
1-07757
609
96
6
950,000
1*08199
+ 167
+ 10
17
3,050,000
1-07564
802
127
6
1,000,000
1-07649
+ 717
+44
16
3,100,000
1-07623
743
120
6
1,050,000
1-08047
+ 319
+ 21
16
3,150,000
1-07623
743
122
6
1,100,000
1-07759
-h 607
+41
15
3,200,000
1-07831
535
89
6
1,150,000
1-07791
+ 575
+40
14
3,250,000
I -080 1 8
348
58
6
1,200,000
1-08628
— 262
-19
14
3,300,000
1-07956
410
70
6
1,250,000
1-08126
+ 240
+ 18
13
3,350,000
1-07956
410
71
6
1,300,000
I -08060
+ 306
+ 24
13
3,400,000
1-07854
512
89
6
i,350i000
I-077S5
+ 611
+49
13
3,450,000
i-o8ii8
248
44
6
1,400,000
1-08302
+ 64
+ 5
12
3,500,000
1-07672
694
124
6
1,450,000
1-07985
+ 381
+32
12
3»55o,ooo
1-07365
lOOI
181
6
1,500,000
I -0806 1
+ 305
+ 27
12
3,600,000
1-07371
995
182
5
i,S5o>ooo
I -0802 1
+ 345
+ 31
II
3,650,000
1-07523
843
156
5
1,600,000
1-07602
+ 764
+ 70
II
3,700,000
1-07661
705
132
5
1,650,000
1-07742
+ 624
+ 59
II
3,750,000
1-07742
624
118
5
1,700,000
1-07940
+ 426
+41
10
3,800,000
1-07576
790
152
5
1,750,000
1-07796
+ 570
+ 56
10
3,850,000
1*07763
603
117
5
1,800,000
1-07707
+ 659
+ 67
10
3,900,000
1-07726
640
126
5
1,850,000
1-07734
+ 632
+66
10
3,950,000
1-07810
556
no
5
1,900,000
1-07982
+ 384
+ 41
9
4,000,000
1-07482
884
177
5
1,950,000
1-08152
+ 214
+ 24
9
4,050,000
1-07761
60s
123
S
2,000,000
1-07971
+ 395
+44
9
4,100,000
1-07754
612
"5
5
2,050,000
1-08032
+ 334
+38
9
4,150,000
1-07528
838
174
5
2,100,000
1-07915
+ 451
+ 52
9
4,200,000
1-07644
+ 722
+ 151
5
78
INTEODTJCnON.
[§ra.
TABT.F. m.
(continued)
•
Difierenoe
Difierence
Difference
Difference
•
between
in A pro-
duoea by
between
in A pzo-
ducedby
number of
number of
1-06366
prim PR and
difference
1-06366
primes and
difference
X,
A.
-A.
number
of lin
X.
A.
-A.
number
of 1 in
given by
number of
•
giyen by
number of
Legendre's
primee
Legeudre'e
primnB
formula.
counted.
formula.
counted.
4,250,000
1*07604
+ 762
+ 161
5
6,650,000
1-07348
+ 1018
+316
3
4,300,000
1*07446
920
196
5
6,700,000
1*07278
1088
340
3
4i35o>ooo
1-07390
976
2IO
5
6,750,000
1*07372
994
3^3
3
4,400,000
1-07305
1061
231
5
6,800,000
1-07328
1040
329
3
4,450,000
1-07597
769
169
5
6,850,000
1-07346
1020
325
3
4,500,000
1-07674
692
154
S
6,900,000
1*07508
858
275
3
4»55o»ooo
1-07464
902
202
4
6,950,000
I-07545
821
265
3
4,600,000
I-07539
827
187
4
7,000,000
I-07559
807
262
3
4,650,000
1-07570
796
182
4
7,050,000
1-07700
666
218
3
4,700,000
1-07580
786
181
4
7,100,000
1*07639
727
239
3
4,750,000
1.07586
780
181
4
7,150,000
1-07592
774
256
3
4,800,000
1*07452
914
214
4
7,200,000
1-07544
822
273
3
4,850,000
1^07482
884
209
4
7,250,000
1-07573
793
265
3
4,900,000
1^07697
669
160
4
7,300,000
I-07593
773
260
3
4,950,000
ro7688
678
163
4
7*350,000
1-07499
867
294
3
5,000,000
1-07836
530
129
4
7,400,000
1-0751S
851
291
3
5,050,000
1-07676
690
169
4
7,450,000
1-07586
780
267
3
5,100,000
1-07719
647
153
4
7,500,000
1-07456
910
314
3
5,150,000
1-07706
660
164
4
7 »5 50,000
1*07450
916
318
3
5,200,000
1-07603
763
192
4
7,600,000
1*07428
938
327
3
5,250,000
1-07783
583
148
4
7,650,000
1-07486
880
308
3
5,300,000
1-07721
645
165
4
7,700,000
1-07533
833
294
3
5,350,000
1-07729
637
164
4
7,750,000
1*07428
938
332
3
5,400,000
I-0774S
621
161
4
7,800,000
1-07365
100 1
357
3
5,450,000
1-07695
671
176
4
7,850,000
1-07559
807
325
3
5,500,000
1-07706
660
174
4
7,900,000
1-07544
822
296
3
5,550,000
I-07547
819
218
4
7,950,000
1-07547
819
297
3
5,600,000
1-07562
804
215
4
8,000,000
1-07487
879
320
3
5,650,000
1-07409
957
258
4
8,050,000
1*07420
946
347
3
5,700,000
1-07456
910
247
4
8,100,000
1*07379
987
364
3
5,750,000
1-07481
885
243
4
8,150,000
1*07336
1030
382
3
5,800,000
1-07319
1047
289
4
8,200,000
1-07423
943
351
3
5,850,000
1-07324
1042
290
4
8,250,000
1*07477
889
333
3
5,900,000
1-07385
981
275
4
8,300,000
1*07431
935
352
3
5,950,000
1-07516
850
240
3
8,350,000
1-07518
848
321
3
6,000,000
1-07418
948
270
3
8,400,000
1-07457
909
346
3
6,050,000
1-07377
989
283
3
8,450,000
1-07341
1025
391
3
6,100,000
1-07396
970
280
3
8,500,000
1-07389
977
375
3
6,150,000
1-07406
960
279
3
8,550,000
1*07458
908
350
3
6,200,000
1-07410
956
280
3
8,600,000
1-07545
821
319
3
6,250,000
1-07314
1052
310
3
8,650,000
1-07538
828
323
3
6,300,000
1-07526
840
249
3
8,700,000
I-07593
773
303
3
6,350,000
1-07518
848
252
3
8,750,000
1*07618
748
295
3
6,400,000
1-07367
999
300
3
8,800,000
1-07698
668
264
3
6,450,000
1-07242
1124
340
3
8,850,000
1-07616
750
285
3
6,500,000
1-07266
1 100
335
3
8,900,000
1*07614
752
301
3
6,550*000
1-07304
1062
326
3
8,950,000
1-07639
727
292
3
6,600,000
1-07366
+ 1000
•+■309
3
9,000,000
1-07666
+ 700
+ 282
2
It will be noticed that the diminution in the value of the constant is very slow, as it only
varies between 1'090 and 1-072 in the whole nine millions. The constant A is very sensitive,
i. e. a small difference in the number of primes counted causes a considerable difference in A ; and
conversely, a small change in the value of A produces but little influence upon the values given
by the formula, especially for the smaller values of x.
The values of A were calculated from the formula A=loga?— tt^, where ^{x) denotes the
number of primes inferior to xi and it may be remarked that A contains one figure less than
1
§ in.] APPEOXIMATIONS TO TCHEBTCHEFE'S FOEMULA, 79
logo? or rpv, a figure being lost by the subtraction. Thus, for example, if a?=:250,000, we have
log a?= 12-42922, ^=11-34044; whence A=12'42922-ll-34044=l-08878 ; so that the
seventh figure in log x or -^r^ influences the sixth figure in the constant.
If n be the number of primes inferior to ar, and if A denote the value of the constant
determined from this number, and A+A denote what would have been its value if the number
of primes counted had been n-^^ instead of n, then
31 St
log«-A— **' log«-A-A— '*^-^'
whence A= /"^t\ . The numbers in the last column of the Table were calculated from the
formula A =15.
It was shown by Tchebycheff, in his Memoir, that if log^""J7^ \iv!i% a limit when x is
infinite, that limit must be unity. It follows therefore that if the number of primes be
Si
represented by a formula of the form ^ _^ , the limiting value of A is unity.
If we determine A so that ^ _^ =li x^ we have A=log ^— if^.
^^"^^g ,. ^ /. . 1 . 2! . 8!
we have
^^^ — loga?\^+logar+(loga?)« + (loga?)»+ *^- /»
g^=loga?(^l+l^+^j^^^+(-j3^^+ &C. j
_1 1 J_ 3 18 71 ^
— ^^8^"'^~loga?"Gogar)«""(logxp)8"'(log^)*~ «:c.;
and therefore 18 18 71
-^=^+i^+(k5g^«+(loga?)8+(log*)4+ *^-
As a first approximation we have -A.=l+j— ^, the value of A being ultimately equal to
^
unitv when x is infinite, as it should be ; and the formula becomes . - 1 *
loga?-l-i—
The series used for li x is semiconvergent, and all the terms have the positive sign ; but the
resulting value of A can be derived without the use of this series from Theorem V. of Tcheby-
cheflTs Memoir. This Theorem is that if the function which denotes the number of primes inferior
Si
to X can be represented algebraically, up to quantities of the order yi r;; inclusive, by means of
the functions or, log ar, ^, it will be represented by the formula
X \,x l.&g 1.2.8...(»-l)jg
loga?"*"(logir)«+(loga?)8*"^ (loga?)» '
The use of this theorem is equivalent to that of the semiconvergent series for li x.
It seemed worth while to calculate the values given by the formulae
X
1 and
for the sake of comparison with those given by Legendre's formula ; the results are shown in the
following table : —
80
INTBODTJCTION.
[§ III.
TABLE IV. — CoMPABisow with
X
X
loga?-108366'
logor-l-i^g^
INTERTAM OP 50,000 UP TO 9,000,000.
r ' ^ isiiiid ^"f
07.
50,000
100,000
150,000
200,000
250,000
300,000
350,000
400,000
450,000
500,000
550,000
600,000
650,000
700,000
750,000
800,000
850,000
900,000
950,000
1,000,000
1,050,000
1,100,000
1,150,000
1,200,000
1,250,000
1,300,000
i,35o»ooo
1,400,000
1,450,000
1,500,000
i,55o»ooo
1,600,000
1,650,000
1,700,000
1,750,000
1,800,000
1,850,000
1,900,000
1,950,000
2,000,000
2,050,000
2,100,000
2,150,000
2,200,000
2,250,000
2,300,000
2,350,000
2,400,000
2,450,000
2,500,000
2,600,000
2,650,000
2,700,000
2,750,000
2,800,000
2,850,000
2,900,000
2,950,000
3,000,000
Ooonted.
5^34
9593
13849
17985
22045
25998
29978
33861
37707
41539
45323
49099
52832
5<5544
60239
63952
67618
71275
74908
78499
82135
85715
89304
92940
96470
10002 I
103543
107124
I 10626
114152
I I 7660
121125
124633
128140
131607
135072
138542
142029
i455o«
148932
152382
155806
159251
162663
166082
169512
172874
176303
179685
183073
186464
189882
193258
196647
199995
203363
206790
210109
213454
216817
Number of Primes.
Oaloulated bj
logJP-1-08366'
5136
9588
13844
17982
22035
26024
29961
33854
37709
41533
45327
49096
52841
56565
60269
63955
67625
71279
74918
78543
82156
85756
89344
92921
96488
100045
103592
I07129
I 10658
H4179
I17691
121195
124692
128181
131663
135139
138608
142070
145526
148976
152420
155858
159291
162718
166140
169557
172969
176376
179778
183175
186568
189956
193340
196720
200095
203467
206834
210197
213557
216913
log*— 1— =
^ log*
5140
9591
13845
17979
22029
26014
29947
33836
37687
41507
452.97
49062
52803
56522
60222
63904
67569
71218
74853
78474
82081
85677
89261
92833
9639s
99947
103490
107023
I 10547
I 14063
117570
121069
124562
128046
131524
134995
138459
141916
145367
148813
152252
155685
159113
162535
165953
169365
172772
176174
179571
182964
186352
189735
193114
196489
199860
203227
206588
209947
213302
216652
logjr-1'
5092
9512
13738
17847
21874
25836
29748
33616
37447
41246
45017
48762
52484
56185
59867
63531
67178
70810
74427
78030
81621
85200
88767
92323
95869
99405
102931
106448
109956
I 13456
I I 6943
120432
123908
127378
130840
134295
137744
141 187
144623
148053
151478
154896
158310
161718
165120
168518
171910
175298
178681
182059
185433
188802
192167
195528
198885
202237
205586
208930
212271
215608
Difference between numbers counted and
cftloulated by
logx-i-ossee*
+
+
+
+
+
+
+
+ 2
- 5
- 5
- 3
- 10
+ 26
- 17
- 7
+ 2
- 6
+ 4
3
9
21
30
3
7
4
10
44
21
41
40
- 19
+ 18
24
49
5
32
27
31
70
59
41
56
67
66
41
24
44
38
52
40
55
58
45
95
73
93
102
104
74
82
73
100
104
44
88
103
4- 96
log*— 1
log*
+ 6
- 2
- 4
- 6
- 16
+ 16
- 31
25
20
32
26
37
29
22
17
48
49
57
55
25
54
38
43
107
75
74
53
lOI
79
89
90
56
71
94
83
77
83
"3
135
119
130
121
138
128
129
147
102
129
114
109
112
147
144
158
135
136
202
162
152
- 165
log*— r
§ III.]
COMPARISON WITH VAEIOUS FORMULA.
81
TABLE lV.(continu.ed).
Number of Primes.
1
Difference between numbers counted and 1
calculated by |
X,
Oounted.
Calculated by
X
X
X
X
X
X
log :p- 108366*
log*— 1— , —
^ logo:
logar— 1*
logx-1-08366'
"^' ' log*
logx-r
3,050,000
220137
220264
219999
218942
+ 127
-138
-"95
3,100,000
223493
223613
223343
222271
120
150
1222
3,150,000
226836
226958
226683
225598
122
153
1238
3,200,000
230210
230299
230019
228920
89
191
1290
3,250,000
233578
233636
233352
232240
58
226
1338
3,300,000
236901
236971
236681
235556
70
220
1345
3»3 50*000
240231
240302
240007
238868
71
224
^3^3
3,400,000
243540
243629
243330
242178
89
210
1362
3,450,000
246910
246954
246650
245484
44
260
1426
3,500,000
250151
250275
249966
248787
124
185
1364
3*550,000
253412
253593
253289
252087
181
123
1325
3,600,000
256726
256908
256589
255384
182
137
1342
3,650,000
260064
260220
259896
258677
156
168
1387
3,700,000
263397
263529
263201
261968
132
196
1429
3,750,000
266717
266835
266502
265256
118
215
1461
3,800,000
269987
270139
269800
268542
152
187
1445
3,850,000
273322
273439
273095
271824
117
227
1498
3,900,000
276611
276737
276388
275103
126
223
1508
3,950,000
279921
280031
279677
278380
no
244
1541
4,000,000
283146
283323
282965
281654
177
181
1492
4,050,000
286490
286613
286249
284926
123
241
1564
4,100,000
. 289774
289899
289531
288195
"5
243
1579
4,150,000
293010
293184
292810
291461
174
200
1549
4,200,000
' 296314
296465
296087
294725
151
227
1589
4,250,000
i 299583
299744
299361
297986
161
222
1597
4,300,000
302824
303020
302632
301244
196
192
1580
4,350,000
306084
306294
305901
304501
210
183
1583
4,400,000
309335
309566
309168
307754
231
167
1581
4,450,000
312666
312835
312432
31 1006
169
234
1660
4,500,000
1 315948
316102
315694
3M255
154
254
1693
4,550,000
1 319164
319366
318953
317502
202
211
1662
4,600,000
322441
32262S
322209
320746
187
232
1695
4,650,000
325706
325888
325465
323988
182
241
1718
4,700,000
328964
329145
328718
327228
181
246
1736
4,750,000
332219
332400
331968
330466
181
251
1753
4,800,000
335439
335653
335216
333701
214
223
1738
4,850,000
338695
338904
338462
336935
209
233
1760
4,900,000
, 341993
342153
341706
340166
160
287
1827
4,950,000
345236
345399
344947
343395
163
289
1841
5,000,000
348515
348644
348186
346622
129
329
1893
5,050,000
351717
351886
351423
349847
169
294
1870
5,100,000
354973
355126
354659
353069
153
314
1904
5,150,000
358200
358364
357892
356290
164
308
1910
5,200,000
361409
361601
361124
359509
192
285
1900
5,250,000
1 364687
364835
364353
362726
148
334
1961
5,300,000
367902
368067
367580
365941
165
322
1961
5,350,000
371133
371297
370805
369154
164
328
1979
5,400,000
374364
374525
374029
372365
161
335
1999
5,450,000
377576
377752
377250
375574
176
326
2002
5,500,000
380802
380976
380469
378781
174
333
2021
5*550,000
383981
384199
383687
381987
218
294
1994
5,600,000
387204
387419
386903
385190
215
301
2014
5,650,000
390380
390638
390117
388392
258
263
1988
5,700,000
393608
393855
393329
391592
247
279
2016
5,750,000
396828
397071
396539
394790
243
289
2038
5,800,000
399995
400284
399748
397986
289
247
2009
5,850,000
403206
403496
402954
401181
290
252
2025
5,900,000
1 406431
406706
406159
404374
27s
272
2057
5,950,000
409674
409914
409362
407565
240
312
2109
6,000,000
412851
413121
412564
410754
+ 270
-287
— 2097
M
82
INTEODTJCTION.
[^ in.
TABLE IV. (continued).
Number of Primes.
1
Difference between numbers counted and 1
calculated by . -1
X,
Counted.
Calculated by
X
*
*
X
*
X
log*- 1-08366'
log*— 1— i
^ log*
lOgJf-l'
log*-l-08366*
^"«' ^ log*
log*—!'
6,050,000
416042
416325
415764
413942
+ 283
— 278
— 2100
6,100,000
419248
419528
418962
4I7I28
280
286
2120
6,150,000
422451
422730
422158
420313
279
293
2138
6,200,000
425650
425930
425353
423496
280
297
2154
6,250,000
428818
429128
428546
426677
310
272
2141
6,300,000
432075
432324
431738
429856
249
337
2219
6,35o»ooo
435267
435519
434928
433034
252
339
2233
6,400,000
438412
438712
4381 16
4362 I I
300
296
2201
6,450,000
441564
441904
441303
439386
340
261
2178
6,500,000
444759
445094
444488
442559
335
271
2200
6,550,000
447957
448283
447672
445731
326
285
2226
6,600,000
451161
451470
450854
448901
309
307
2260
6,650,000
454339
454655
454034
452070
316
305
2269
6,700,000
457499
457839
457213
455237
340
286
2262
6,750,000
460709
461022
460391
458402
313
318
2307
6,800,000
463874
464203
463567
461567
329
307
2307
6,850,000
467057
467382
466741
464729
325
316
2328
6,900,000
470285
470560
469914
467891
275
371
2394
6,950,000
473472
473737
473085
47I05I
265
387
2421
7,000,000
476650
476912
476256
474209
262
394
2441
7,050,000
479868
480086
479425
477366
218
443
2502
7,100,000
483019
483258
482592
480522
239
426
2497
7,150,000
486173
486429
485758
483676
256
415
2497
7,200,000
4B9325
489598
488922
486829
273
403
2496
7,250,000
492501
492766
492086
489980
265
415
2521
7,300,000
495673
495933
495247
493130
260
426
2543
7>3So»ooo
498805
499099
498408
496279
294
397
2526
7,400,000
501972
502263
501567
499427
291
405
2545
7,450,000
505158
505425
504724
502573
267
434
2585
7,500,000
508273
508587
507881
505718
314
392
2555
7*550,000
511429
5"747
51 1036
508861
318
393
2568
7,600,000
514578
514905
514189
512003
327
389
2575
7,650,000
517754
518062
517342
5I5I44
308
412
2610
7,700,000
520925
521219
520493
518284
294
432
2641
7,750,000
524041
524373
523643
521422
332
398
2619
7,800,000
527170
527527
526792
524559
357
378
261 1
7,850,000
530354
530679
529939
527695
325
415
2659
7,900,000
533534
533830
533084
530829
296
450
2705
7,950,000
536683
536980
536229
533963
297
454
2720
8,000,000
539808
540128
539373
537095
320
435
2713
8,050,000
542929
543276
542515
540226
347
414
2703
8,100,000
546058
546422
545656
543355
364
402
2703
8,150,000
549185
549567
548796
546484
382
389
2701
8,200,000
552359
552710
551935
54961 I
351
424
2748
8,250,000
555520
555853
555072
552737
333
448
2783
8,300,000
558642
558994
558208
555862
352
434
2780
8,350,000
561813
562134
561343
558986
321
470
2827
8,400,000
564927
565273
564477
562108
346
450
2819
8,450,000
568019
568410
567609
565230
391
410
2789
8,500,000
571172
571547
570742
568350
375
430
2822
8,550,000
574332
574682
573873
571469
350
459
2863
8,600,000
577498
577817
577003
574587
319
495
2911
8,650,000
580627
580950
580129
577704
323
498
2923
8,700,000
583779
584082
583257
580820
303
522
2959
8,750,000
586918
587213
586383
583934
29s
535
2984
8,800,000
590078
590342
589507
587048
264
571
3030
8,850,000
593186
593471
592631
590160
285
555
3026
8,900,000
596298
596599
595753
593272
301
545
3026
8,950,000
599433
599725
598875
596382
292
558
3051
9,000,000
602568
602850
601995
599491
+ 282
-573
-3077
§ m.] BIEMANN'S FOEMULA. 88
It will be noticed that, for the values of x contained in the table, the formula
^
loga;— 1 —
log^
is not so satisfactory as Legendre's. The values given by the formula ^^ _^ are very wide of
the truth,
Riemann's Formula.
In 1859 Biemann communicated to the Berlin Academy a paper, " Ueber die Anzahl der
Primzahlen unter einer gegebenen Grosse," which occupies pp. 136-144 of his collected works.
He shows that, if ^(ar) denotes the number of primes inferior to a?, then ^{x) consists of a non-
periodic function and a series of periodic terms involving the roots of a certain transcendental
equation* The non-periodic function is
li(ar)-iU(a:*)-iU(a:*)-lU(a;^)+ili(4?*)---^li(a:*)+Ali(^^^^^
where the terms are of the form ±~ li C^)? n being a product of prime factors of the form
ahc . . . {&o that n contains no squared factor), and the sign being positive when the number of
factors a, i, c, . . . is even, and negative when it is uneven. We may therefore represent the
number of primes inferior to x approximately by the formula
li ^— ^ li ar^— ^ li ^* — ^ li 0?*+^ li x^—Scc.
As this formula is the only one that can be regarded as founded upon a rigorous investi-
gation, it seemed very desirable to compare the values given by it with the numbers of primes
counted, although the calculation of the formula is troublesome, both on account of the form of
the arguments and the slow convergence of the terms.
I accordingly calculated the values of the formula for the same values of ^ as in the case of
the other formulae, and the results are contained in Table V. The corresponding values given
by Legendre's and TchebycheflTs formulse are also added for the sake of comparison.
The great superiority of Eiemann's formula to the others is at once apparent ; it is more
accurate than Legendre's even for the smaller values of Xy and it represents the numbers of primes
over the whole nine millions most satis£ax)torily. It scarcely seems possible that a continuous
formula not involving periodic terms could more accurately represent numbers which exhibit such
great irregularities.
Table VI. exhibits the deviations between the numbers of primes counted and the numbers
given by the five formulae : —
(i.) Biemann's, lia?— ^li**— . . .
(ii.) Tchebycheffa or Gauss's, li x.
(iii.) Legendre's, ioga;-l08366 -
(iv.)
logo?— 1
(V.)
logo?
X
logo?— l'
m2
84
INTBODTTCTION.
tt "I-
TABLE V. — Comparison with Ribmann's, Tchbbtchbfp's, and Leoendse's FoBMULiS; at intebtaui
OP 50,000 UP TO 9,000,000.
Number of Primes
DiiFerenoe between numbers counted and 1
calculated by |
X,
Oounted.
Calculated by
Hjr-iUor*-...
Ux.
«
li«-iUx*-...
U«.
X
log«-l-08366"
logx-i-oeaee'
50,000
5134
5133
5167
5136
— I
+ 33
+ 2
100,000
9593
9587
9630
9588
- 6
37
- 5
150,000
13849
13844
13893
13844
- 5
44
- 5
200,000
17985
17982
18036
17982
- 3
51
- 3
250,000
22045
22035
22094
22035
— 10
49
— 10
300,000
25998
26024
26087
26024
+ 26
89
+ 26
350,000
29978
29960
30027
29961
- 18
49
- 17
400,000
33861
33852
33923
33854
- 9
62
- 7
450,000
37707
37707
37781
37709
74
+ 2
500,000
41539
41530
41606
41533
- 9
67
- 6
550,000
45323
45323
4^403
45327
80
+ 4
600,000
49099
49091
49173
49096
- 8
74
- 3
650,000
52832
52835
52920
52841
+ 3
88
t 9
700,000
5^5544
56557
56645
56565
4- 13
lOI
+ 21
750,000
60239
60261
60350
60269
-h 22
III
+ 30
800,000
63952
63945
64037
63955
- 7
85
+ 3
850,000
67618
67614
67708
67625
- 5
90
i '
900,000
71275
71266
71362
71279
- 9
87
+ 4
950,000
74908
74904
75002
74918
- 4
94
+ 10
1,000,000
78499
78528
78628
78543
+ 29
129
+ 44
1,050,000
82135
82138
82240
82156
+ 3
105
+ 21
1,100,000
8571S
85737
85841
85756
+ 22
126
+ 41
1,150,000
89304
89323
89429
89344
+ 19
125
+ 40
1,200,000
92940
92899
93007
92921
— 41
67
7 ^9
1,250,000
96470
96464
96573
96488 ,
- 6
103
+ 18
1,300,000
10002 1
100019
100130
100045
— 2
109
24
i»35o»ooo
103543
103564
103677
103592
+ 21
134
49
1,400,000
107124
107 100
107214
107129
- 24
90
5
1,450,000
I 10626
I10627
I 10743
I 10658
+ I
117
32
1,500,000
114152
I14146
I 14263
114179
- 6
III
27
i»S5o>ooo
I I 7660
I17656
"7775
117691
- 4
"5
31
1,600,000
121125
121159
121279
121195
+ 34
154
70
1,650,000
124633
124653
124775
124692
+ 20
142
59
1,700,000
128140
128141
128264
128181
+ I
124
41
1,750,000
131607
131621
131746
131663
+ 14
139
56
1,800,000
135072
135095
135221
135139
+ 23
149
67
1,850,000
138542
138561
138689
138608
+ 19
147
66
1,900,000
142029
142022
142150
142070
- 7
121
41
1,950,000
145502
145476
145606
145526
- 26
104
24
2,000,000
148932
148924
149055
148976
- 9
123
44
2,050,000
152382
152366
152498
152420
- 17
116
38
2,100,000
155806
155802
155936
155858
— 4
130
52
2,150,000
159251
159232
159368
159291
- 19
117
40
2,200,000
162663
162658
162794
162718
- 6
^31
55
2,250,000
166082
166077
166215
166140
- 5
^33
58
2,300,000
169512
169492
169631
169557
— 20
119
45
2»35o>ooo
172874
172902
173042
172969
+ 28
168
95
2,400,000
176303
176307
176448
176376
+ 4
145
73
2,450,000
179685
179706
179849
179778
+ 21
164
93
2,500,000
183073
183102
183245
183175
+ 29
172
102
2,550*000
186464
186492
186637
186568
+ 28
173
104
2,600,000
189882
189878
190024
189956
- 4
142
74
2,650,000
193258
193260
193407
193340
+ 2
149
82
2,700,000
196647
196637
196785
196720
— 10
138 *
73
2,750,000
199995
20001 I
200160
200095
+ 16
165
100
2,800,000
203363
203380
203530
203467
+ 17
167
104
2,850,000
206790
206745
206896
206834
- 45
106
44
2,900,000
210109
210106
210258
210197
- 3
149
88
2,950,000
213454
213463
213616
213557
+ 9
162
103
3,000,000
216817
216816
216971
216913
— I
+ 154
+ 96
§ III.]
EIEMANN'S, TCHEBTCHEFF'S, AND LEGENDRE'S FOEMULiE.
85
TABLE V. (continued).
Number of Primefl.
1
DifTerenoe between numbers counted and 1
calculated by |
X,
Counted.
Calculated bj
li*— ^lijr— ...
U«.
X
liar— ilijr— ...
hx.
X
.^^
log ar~ 108366'
log;r-l-08366*
3,050,000
220137
220166
220321
220264
+ 29
+ 184
+ 127
J,joo,ooo
223493
223512
223668
223613
-I- 19
175
120
3,150,000
226836
226854
227012
226958
+ 18
176
122
J,20<=>,000
230210
230193
230351
230299
- 17
141
89
3,2 s <:>,ooo
233578 .
233529
233688
233636
- 49
no
58
3»3»^^»^°°
236901
236961
237021
236971
- 41
120
70
3,3S*=»>ooo
240231
240189
240350
240302
- 42
119
71
3,4.0 ^=>,ooo
243540
243514
243677
243629
- 26
137
89
3,4.5 o=>,ooo
246910
246837
247000
246954
- 74
90
44
3,50 4=^,000
250151
250155
250319
250275
+ 4
168
124
3,55<z^,ooo
253412
253471
253636
253593
+ 59
224
181
3,6o<:3,ooo
256726
256784
256950
256908
+ 58
224
182
3,654=3,000
260064
260093
260260
260220
+ 29
196
156
3,7o<=>,ooo
263397
263400
263568
263529
+ 3
171
132
3.7 5 *a,ooo
266717
266704
266872
266835
- 13
155
118
3,8o4=3,ooo
269987
270004
270174
270139
+ 17
186
152
3,854=3,000
273322
273302
273473
273439
— 20
151
117
3,90 4=»,ooo
276611
276597
276769
276737
- 14
158
126
3,95 <=>,ooo
279921
279890
280062
280031
- 31
141
no
4,ocx3,ooo
283146
283179
283352
283323
+ 33
206
177
4,05 ^>,ooo
286490
286466
286640
286613
— 24
150
123
4,100,000
289774
289750
289925
289899
- 24
151
125
4,1503,000
293010
293032
293208
293184
4- 22
198
174
4,20<:>,ooo
296314
29631 I
296487
296465
- 3
173
151
4f25<i>,ooo
299583
299587
299765
299744
+ 4
182
161
4930<^,ooo
302824
302861
303039
303020
+ 37
215
196
4,35^3,000
306084
306133
306312
306294
+ 49
228
210
4,4oc3,ooo
309335
309402
309582
309566
+ 67
247
231
4,45 <3,0OO
312666
312668
312849
312835
+ 2
183
169
4»5<^<3,ooo
315948
315933
316114
316102
- 16
166
154
4>55<^,ooo
319164
319194
319377
319366
+ 30
213
202
4»oo^>ooo
322441
322454
322637
322628
+ 13
196
187
4,050,000
325706
3257"
325895
325888
+ 5
189
182
4,7oc3,ooo
328964
328965
329150
329145
+ 2
186
181
4»75c3,ooo
332219
332218
332404
332400
— I
185
181
4,»0c3,000
335439
335469
335655
335653
+ 30
216
214
4>S5c3,ooo
338695
338717
338904
338904
+ 22
209
209
4,90^3,000
341993
341963
342151
342153
- 30
158
160
4»S>5c^,ooo
345236
345207
345396
345399
- 29
160
163
^>^^>c^,ooo
348515
348449
348638
348644
- 66
123
129
169
5»^ 5 0,000
351717
351689
351879
351886
- 29
162
5> 1 00,000
354973
354926
35S"7
355126
- 47
144
153
5»^S 0,000
358200
358162
358353
358364
- 38
153
164
5*300,000
361409
361395
361588
361601
— 14
179
192
5>^ 5 0,000
364687
364627
364820
364835
- 60
^33
148
5>3c>c>,ooo
ET -■_ '
367902
367856
368050
368067
- 46
148
1^5
*^»3 50,000
371133.
371084
371279
371297
- 49
146
164
5>-«1.oo,ooo
374364
374310
374505
374525
- 55
141
161
5»-^5 0,000
377576
377533
377729
377752
- 43
153
176
5>S«io,ooo
380802
380755
380952
380976
- 47
150
174
^» 5 50,000
383981
383975
384173
384199
- 6
192
218
o>0c>o,000
ET ^T
387204
387193
387391
387419
-— II
187
215
^*^ 50,000
390380
390409
390608
390638
+ 29
228
258
5>V 00,000
I? — '
393608
393624
393823
393855
+ 16
215
247
^^v 50,000
396828
396836
397037
397071
+ 8
209
243
5 > ^00,000
399995
400047
400248
400284
+ 52
253
289
^*^ 50,000
403206
403256
403458
403496
+ 50
252
290
S-fc^OOjOOO
406431
406463
406666
406706
+ 32
235
275
^» 950,000
^'»000,000
409674
409669
409872
409914
- S
198
240
412851
412873
413077
413121
+ 22
+ 226
+ 270
86
INTEODUCTION.
[§ ni.
TABLE V. (continued).
Number of PrimeB.
Difference between numbers counted and 1
calculated bj |
X,
Counted.
Calculated by
Ii«— Jlijr— ...
lix.
X
liar— ilijr— ...
lix.
X
logx-i-ossee*
log*- 108366*
6,050,000
416042
416075
416279
416325
+ 33
+ 237
+ 283
6,100,000
419248
41927s
419480
419528
+ 27
232
280
6,150,000
422451
422474
422680
422730-
-h 23
229
279
6,200,000
425650
425671
425878
425930
+ 21
228
280
6,250,000
428818
428866
429074
429128
•f 48
256
310
6,300,000
432075
432060
432268
432324
- 15
193
249
6,3S^,ooo
435267
435253
435461
435519
- 15
194
252
6^400,000
438412
438443
438652
438712
+ 31
240
300
6,450,000
441564
441632
441842
441904
-f 68
278
340
6,500,000
444759
444819
445030
445094
-f 60
271
335
6,550*000
447957
448005
448216
448283
+ 48
259
326
6,600,000
451161
451190
451401
451470
+ 29
240
309
6,650,000
454339
454372
454585
454655
+ 33
246
316
6,700,000
457499
457554
457767
457839
•f 55
268
340
6,750,000
460709
460733
460947
461022
+ 24
238
3^3
6,800,000
463874
463912
464126
464203
+ 38
252
329
6,850,000
467057
467088
467303
467382
+ 31
246
325
6,900,000
470285
470263
470479
470560
— 21
194
275
6,950,000
473472
473438
473654
473737
- 35
182
265
7,000,000
476650
476610
476827
476912
- 40
177
262
7,050,000
479868
479781
479998
480086
- 87
130
218
7,100,000
483019
482950
483169
483258
- 69
150
239
7,150,000
486173
4861 19
486337
486429
- 55
164
256
7,200,000
489325
489285
489505
489598
- 40
180
273
7,250,000
492501
492451
492671
492766
- 51
170
265
7,300,000
495673
495615
495835
495933
- 59
162
260
7^350.000
498805
498777
498998
499099
- 28
193
294
7,400,000
501972
501938
502160
502263
- 34
188
291
7,450,000
505158
505098
505321
505425
- 60
163
267
7,500,000
508273
508257
508480
508587
- 16
207
314
7,550,000
511429
511414
511638
511747
- 15
209
318
7,600,000
514578
514570
514794
514905
- 8
216
327
7,650,000
517754
517724
517949
518062
- 30
195
308
7,700,000 i
520925
520878
521103
521219
- 47
178
294
7,750,000
524041
524030
524256
524373
— II
215
332
7,800,000
527170
527180
527407
527527
+ 10
237
357
7,850,000
530354
530330
530557
530679
- 24
203
325
7,900,000
533534
533478
533706
533830
- 56
172
296
7,950,000
536683
536625
536853
536980
- 58
170
297
8,000,000
539808
539771
540000
540128
- 37
192
320
8,050,000
542929
542915
543145
543276
— 14
216
347
8,100,000
546058
546058
546289
546422
231
364
8,150,000
549185
549200
549431
549567
+ 15
246
382
8,200,000
552359
552341
552573
552710
— 18
214
351
8,250,000
555520
555481
555713
555853
- 39
193
333
* 8,300,000
558642
558619
558852
•558994
- 23
210
352
8,350,000
561813
561756
561990
562134
- 57
177
321
8,400,000
564927
564892
565126
565273
- 35
199
346
8,450,000
568019
568027
568262
568410
+ 8
243
391
8,500,000
571172
571161
571396
571547
■— II
224
375
8,550,000
574332
574294
574529
574682
- 39
197
350
8,600,000
577498
577425
577661
577817
- 73
163
319
8,650,000
580627
580555
580792
580950
- 72
165
323
8,700,000
583779
583684
583921
584082
- 95
142
303
8,750,060
586918
586812
587050
587213
— 106
132
29s
8,800,000
590078
•589939
590178
590342
-139
100
264
8,850,000
593186
593065
593304
593471
— 121
118
285
8,900,000
596298
596190
596429
596599
— 108
131
301
8,950,000
599433
599313
599553
599725
— 120
120
292
9,000,000
602568
602436
602676
602850
— 132
+ 108
+282
§ni.]
DEVIATIONS OF THE FIVE FOEMULiE.
87
TABLE VI.
Devutions op the Five Fo&uvlm.
X,
Biemann.
(ii.)
Tohebjr-
cheff.
(m.)
Legendre.
(iy.)
X
X
X.
(i-)
Biemann.
(ii.)
Tcheby-
cheff.
(iii.)
Legendre.
(iv.)
X
logx-l-. ^
log*
log«-l
log*-l- ^
log*
logjp-1*
50,000
— I
+ 33
+ 2
+ 6
- 42
3,050,000
+ 29
+ 184
+ 127
-138
-"95
100,000
- 6
37
- S
— 2
81
3,100,000
+ 19
175
120
150
1222
150,000
- 5
44
- S
- 4
III
3,150,000
+ 18
176
122
153
1238
200,000
- 3
SI
- 3
- 6
138
3,200,000
-17
141
89
191
1290
250,000
— 10
49
— 10
- 16
171
3,250,000
-49
no
58
226
1338
300,000
+ 26
89
+ 26
+ 16
162
3,300,000
-41
120
70
220
1345
350,000
-18
49
- 17
- 31
230
3»35o,ooo
-42
119
71
224
1363
400,000
- 9
62
- 7
25
' 245
3,400,000
-26
137
89
210
1362
450,000
74
+ 2
20
260
3,450,000
-74
90
44
260
1426
500,000
- 9
67
- 6
32
293.
3,500,000
+ 4
168
124
185
I3<54
550,000
80
+ 4
26
306
3,550,000
+ 59
224
181
123
1325
600,000
- 8
74
- 3
37
337
3,600,000
+ 58
224
182
137
1342
650,000
+ 3
88
+ 9
29
348
3,650,000
+ 29
196
156
168
1387
700,000
+ 13
lOI
+ 21
22
359
3,700,000
+ 3
171
132
196
1429
750,000
+ 22
III
+ 30
17
372
3,750,000
-13
155
118
215
1461
800,000
- 7
85
+ 3
48
421
3,800,000
+ 17
186
152
187
1445
850,000
- 5
90
+ 7
49
440
3,850,000
—20
151
117
227
1498
900,000
- 9
87
+ 4
57
465
3,900,000
-14
158
126
223
1508
950,000
- 4
94
+ 10
55
481
3,950,000
-31
141
no
244
1541
I>000,000
+29
129
+ 44
25
469
4,000,000
+33
206
177
181
1492
19050,000
+ 3
105
+ 21
54
514
4,050,000
-24
150
123
241
1564
1,100,000
+ 22
126
+ 41
38
S15
4,100,000
-24
151
125
243
1579
1,150,000
+ 19
125
+ 40
43
537
4,150,000
+ 22
198
174
200
1549
1,200,000
-41
67
- 19
107
617
4,200,000
- 3
173
151
227
1589
1,250,000
- 6
103
+ 18
75
601
4,250,000
+ 4
182
161
222
1597
1,300,000
— 2
109
24
74
616
4,300,000
+37
215
196
192
1580
ii3SOiOOO
+ 21
134
49
53
612
4,350,000
+ 49
228
210
183
1583
1,400,000
-24
90
5
lOI
676
4,400,000
+ 67
247
231
167
1581.
i>45o>ooo
+ I
117
32
79
670
4,450,000
+ 2
183
169
234
1660
1,500,000
- 6
III
27
89
696
4,500,000
-16
166
154
254
1693
I »S 50*000
- 4
"5
31
90
712
4,550,000
+ 30
213
202
211
1662
1,600,000
+34
154
70
56
693
4,600,000
+ 13
196
187
232
1695
1,650,000
+ 20
142
59
71
725
4,650,000
+ 5
189
182
241
1718
1,700,000
-f I
124
41
94
762
4,700,000
+ 2
186
181
246
1736
I »7 50,000
+ 14
139
56
83
767
4,750,000
_ I
185
181
251
1753
1,800,000
+ 23
149
67
77
777
4,800,000
+ 30
216
214
223
1738
1,850,000
+ 19
147
66
83
798
4,850,000
+ 22
209
209
233
1760
1,900,000
- 7
121
41
"3
842
4,900,000
-30
158
160
287
1827
1,950,000
-26
104
24
135
879
4,950,000
-29
160
163
289
1841
2,000,000
- 9
123
44
119
879
5,000,000
^66
123
129
329
1893
2,050,000
-17
116
38
130
904
5,050,000
-29
162
169
294
1870
2,100,000
- 4
130
52
121
910
5,100,000
-47
144
153
314
1904
2,150,000
-19
117
40
138
941
5,150,000
-38
153
164
308
1910
2,200,000
— 6
131
55
128
945
5,200,000
-14
179
192
285
1900
2,250,000
- S
^33
58
129
962
5,250,000
-60
^33
148
334
1961
2,300,000
—20
119
45
M7
994
5,300,000
-46
148
165
322
' 1961
2»35o,ooo
+ 28
168
95
102
964
5,350,000
-49
146
164
328
1979
2,400,000
+ 4
145
73
129
1005
5,400,000
-55
141
161
335
1999
2,450,000
+ 21
164
9S
114
1004
5,450,000
-43
153
176
326
2002
2,500,000
+ 29
172
102
109
1014
5,500,000
-47
150
174
333
2021
2»55o,ooo
+ 28
173
104
112
103 1
5.550,000
- 6
192
218
294
1994
2,600,000
- 4
142
74
147
1080
5,600,000
— II
187
215
301
2014
2,650,000
+ 2
149
82
144
109 1
5,650,000
+ 29
228
258
263
1988
2,700,000
— 10
138
73
158
1119
5,700,000
+ 16
215
247
279
2016
2,750,000
+ 16
165
100
»35
mo
5,750,000
+ 8
209
243
289
2038
2,800,000
+ 17
167
104
136
1126
5,800,000
+ 52
253
289
247
2009
2,850,000
-45
106
44
202
1204
5,850,000
+ 50
252
290
252
2025
2,900,000
- 3
149
88
162
1179
5,900,000
+ 32
235
275
272
2057
2,950,000
+ 9
162
103
152
1183
5,950,000
- 5
198
240
312
2109
I 3,000,000
— I
+ 154
+ 96
-16s
— 1209
6,000,000
+ 22
+ 226
+ 270
-287
— 2097
88
INTEODUCTIOK.
L^m.
TABLE VI. (continued).
X.
(i.)
Biemann.
(u.)
Tchebjr-
obefi.
(m.)
Legendre.
Civ.)
X
(v.)
X
X.
(i.)
Biemann.
Tohebj-
oheff.
(iii.)
Legendre.
(iv.)
X
(V.)
X
log»-l-, ^
logo?
log«-l
log«-l- 1
log*
logjr-1
6,050,000
+ 33
+ 237
+ 283
-278
— 2100
7,550,000
- 15
+ 209
+ 318
-393
-2568
6,100,000
+ 27
232
280
286
2120
7,600,000
-. 8
216
327
389
2575
6,150,000
+ 23
229
279
293
2138
7,650,000
- 30
195
308
412
2610
6,200,000
+ 21
228
280
297
2154
7,700,000
- 47
178
294
432
2641
6,250,000
+ 48
256
310
272
2I4I
7,750,000
— II
215
332
398
3619
6,300,000
-15
193
249
337
2219
7,800,000
+ 10
237
357
378
2611
6j3SOi^<>
-15
194
252
339
2233
7,850,000
- 24
203
325
415
2659
6,400,000
+ 31
240
300
296
2201
7,900,000
- 56
172
296
450
2705
6,450,000
+ 68
278
340
261
2178
7,950,000
- 58
170
297
454
2720
6,500,000
+ 60
271
335
271
2200
8,000,000
- 31
192
320
435
2713
6,550,000
+ 48
259
326
285
2226
8,050,000
- 14
216
347
414
2703
6,600,000
+ 29
240
309
307
2260
8,100,000
231
364
402
2703
6,650,000
+33
246
316
305
2269
8,150,000
+ IS
246
382
389
2701
6,700,000
+ 55
268
340
286
2262
8,200,000
- 18
214
351
424
2748
6,750,000
+ 24
238
3^3
318
2307
8,250,000
- 39
193
333
448
2783
6,800,000
+38
252
329
307
2307
8,300,000
- 23
210
352
434
2780
6,850,000
+31
246
32s
316
2328
8,350,000
- 57
177
321
470
2827
6,900,000
—21
194
275
371
2394
8,400,000
- 35
199
346
450
2819
6,950,000
-35
182
265
387
2421
8,450,000
+ 8
243
391
410
2789
7,000,000
—40
177
262
394
2441
8,500,000
— II
224
375
430
2822
7,050,000
-87
130
218
443
2502
8,550,000
- 39
197
350
459
2863
7,100,000
-69
150
239
426
2497
8,600,000
- 73
163
319
495
291 1
7.150,000
-55
164
256
415
2497
8,650,000
- 72
165
323
498
2923
7,200,000
-40
180
273
403
2496
8,700,000
- 95
142
303
522
2959
7,250,000
-51
170
265
415
2521
8,750,000
— 106
132
295
535
2984
7,300,000
-59
162
260
426
2543
8,800,000
-139
100
264
571
3030
7f3S^,ooo
-28
193
294
397
2526
8,850,000
— 121
118
285
555
3026
7,400,000
-34
188
291
405
2545
8,900,000
— 108
131
301
545
3026
7,450,000
-60
163
267
434
2585
8,950,000
— 120
120
292
558
3051
7,500,000
-16
+ 207
+314
-392
-2555
9,000,000
— 132
+ 108
+282
-573
-3077
The mean deviations for the five formulse, derived from this table, are respectively
-9, +163. +171, -242, -1643.
The diagram facing this page was constructed in order to exhibit to the eye with greater
distinctness the deviations of the formulsB throughout the extent of the table. Each of the
points laid down was determined as the mean of three of the values given in the table ; thus the
means of the numbers in the first three lines in the table are
-4, +38,
-3,
0,
-78.
and these are taken as the ordinates of the points corresponding to ^=100,000; the means of
the numbers in the next three lines are
+4, +63,
+4,
-2,
-157,
and these are taken as the ordinates of the points corresponding to ^=250,000, and so on.
The diagram shows clearly the excellence of Riemann's formula, the deviations being always
small in amount and changing sign frequently. It may be remarked that, taking Meissel's
value of the number of primes inferior to 10,000,000, which is given on the next page, the
deviation at a;=10,000,000 is +87, so that between 9,000,000 and 10,000,000 Riemann's
deviation curve crosses the zero line again.
The point of intersection of Legendre's and Tchebycheff's formulae at a:= 4,850,000 will also
be noticed. Beyond this point Legendre's formula steadily diverges.
S S § 1 -1 1 1 i i 1 8 S
§ s
1 + OEVIAYIbf
t <. t % I I I I \
^
Jk
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vr
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D£VIATIOI
§ III.]
ON THE CALCULATION OF THE lUNCTION liar.
89
Number of Primes inferiw to 10,000,000 and to 100,000,000.
In the second and third volumes of the Mafhematiache Annalen (1870 and 1871) Meissel has
determined the numbers of primes inferior to 10,000,000 and to 100,000,000 by a method which
is equivalent to actually counting them *, so that his numbers should be exact. Hargreave also,
in the Philosophical Magazine for 1854, had obtained by means of a similar process the number
of primes inferior to 10,000,000. In the case of 10,000,000 Meissel's number is 664,580, and
Hargreave's 664,633 ; for 100,000,000 Meissel's number is 5,761,461. Meissel is so accurate a
calculator that his results are entitled to be accepted with confidence ; and taking his numbers to
represent the actual numbers of primes counted, we obtain the following results : —
X,
Number of PrimeB.
•
Counted by
Calculated by
MeiaeeL
Biemann.
TohebyoheflP.
Legendre.
X
X
^ log*
log*—!'
10,000,000
100,000,000
664,580
5»76i,46i
664,667
5»76i,5Sl
664,918
5,762,209
665,140
5,768,004
664,185
5*758,248
661,459
51740,304
X,
Deyiations of the fire formulflB from the number of primes counted by Meuoel.
Riemann.
Tchebycheff.
Legendre.
X
X
log*— 1— ,
» logx
logx-r
10,000,000
100,000,000
+ 87
+ 90
+ 338
+ 748
+ 560
+ 6543
- 395
-3213
— 3121
-21157
The great accuracy with which Riemann s formula represents the number of primes both
at 10,000,000 and 100,000,000 is very remarkable. At 100,000,000 the function li x still affords
a good approximation, and its superiority to Legendre's formula, which gives a result differing
widely from the truth, is very apparent ; it is noticeable also that the deviation in the case of the
formula
X
log a? — 1 —
y- is less than one half of that shown by Legendre's formula.
logo?
The value of the constant A determined (as in Table III.) from ar=10,000,000 is 1-07110,
and the value determined from ar=100,000,000 is 1*06397. It thus appears that the approach
of the constant towards unity is extremely slow.
On the Calculation of the Function li x.
The logarithm-integral liar, of which so much use has been made in this section, and
which is so closely connected with the average distribution of primes, seems to have been first
considered by Mascheroni f, although the importance of the function had been previously recog-
nized byEuler J. In 1809 Soldner published at Munich a work devoted to its theory, and which
* See Introduction to the Fourth Million, pp. 34, 35.
t See Ahhandlungen von Friedrich Wilhelm Bessel^ vol. ii. (1876), p. 331, or Lacroix, Traite des differences et des
series, § 1117.
singularem spociem functionum transcendentium suppeditare, quae accuratiorem
Tdz
J log 2
evolutionem meretur" (Institutiones Calc. Integ. vol. i. Cap. iv. § 219).
N
90 INTEODUCTION. [§ III.
also contained a table of its values. In Zach's Monatliche Correspondenz for 1812 (vol. xxvi.)
there is a paper by Buzengeiger upon Soldner's table and methods ; and in the same year Bessel
published in the Konigsberger Archiv ftir Naturwissenschafften und MathematiJc some investi-
gations, both analytical and numerical, relating to the function \\x^ which are reprinted in
vol. ii. pp. 330-342 of his collected works.
i
integral is known by the name of the Exponential-integral, and is denoted by Eio:, so that
li ^=Ei X and li a?=Ei(loga:).
The functions \\x and Ei x may, however, be defined more conveniently by the formulae
T 11/1 \ I 1 I Oog^)* I (log^/* , p
h a;=y+log(log ar)+log ar+-^J^ +^^3^+&c.,
Eiar=y+log^+a;+^+3T3+|5+^+&c.
In these series, which are always convergent, y denotes Euler's constant 0*57721566 . . . and
the terms log (log x) and log x^ which occur in the series for li x and Ei x respectively, are to be
understood to denote \ log (log xy and ^ log {a^) ; i. e. it is supposed that log (log x) and log x
are not altered by a change of sign of log x and x respectively.
The integrals li x and Ei x contain infinite elements corresponding to x=l and ar=0 respec-
tively ; if the integrals be regarded as defining the functions, they are supposed to have their
principal values assigned to them.
The exponential-integral is one of a group of three integrals which may be defined by the
equations
Siar=l -^aw, Ci;r=l "^dUy £i^=l — «w,
the two former of which are termed the Sine-integral and the Cosine-integral. Changing the sign
of X we find
Ei(-a?)=i —du.
The functions Si ar, Ci ar, Ei x have been used by Schlomilch, Amdt, and other writers as a
means of expressing the values of more complicated integrals, and they are regarded as known
functions.
The following is, I believe, a complete list of the tables that have been published of the
functions li or, Ei {x\ Ei {—x\ Si or, and Cix: —
(i) 1809. SoLDNER. Theorie et tables d*une nouvelle fonction transcendante.
Table of liar from a;=0 to a;=l at intervals of 001, from ar=l to ^=2 at intervals of 0-1,
and thence at gradually increasing intervals to ar==1280. Throughout most of the table the
number of decimals is seven, but in the latter part the number is six or five. The table is
reprinted in De Morgan's ' Differential and Integral Calculus,' pp. 662-3.
(ii) 1812. Bessel. Untersuchung der durch das Integral j . ausgedriickten trans-
cendenten Function. Abhandlungen, vol. ii. p. 339.
Values of li 1000, li 10,000, 11100,000, 11200,000, 11300,000, U 400,000, U 500,000,
li 600,000, and li 1,000,000 to six or five places of decimals.
§ III.] ON THE CALCULATION OF THE FUNCTION liar. 91
(iii) 1843. Bbetsghneideb. ^^ Berechnung der Grundzahl der natiirlichen Logarithmen,
80 wie mehrerer anderer mit ihr zusammenhangender Zahlwerthe." Grunerfs Arckiv
der Mathematik und Physik^ vol. iii. pp. 27-34.
Values of Si or, Ci or, Ei or, Ei ( —x) for a;=l, 2, 3, ... 10 to twenty places of decimals. In
the case of x=\ the values are extended to thirty-five places. The tahle is reprinted by Schlomilch
at the end of his " Analytische Studien."
(iv) 1861. Bretschneidek. "Ueber die Berechnung der Integrallogarithmen und
eniger mit ihm zusammenhangenden Functionen." Zeitachrift fur Mathematik und
Physik, vol. vi. pp. 127-139.
Values of Siar, Ciar, Eia:, Ei (—!,•) for ar=l, 2, ... 10 to twenty places of decimals, from
x=0 to ar=l at intervals of 0*01, and from a?=l to x=z7'5 at intervals of 0*1 to ten places.
(v) 1861-7. Stenberg. "Tabulse logarithmi-integralis." Malmogiae, Pars I. 1861,
Pars II. 1867.
Table of li 10* from x= — 15 to ar= — 10 at intervals of 0-05, from ar= — 10 to ar=— 7 at
intervals of 002, and thence to j?=+3-5 at intervals of 0*01. As far as x=zO eighteen places
are given, but for positive arguments the number is fifteen, fourteen, or thirteen.
(vi) 1870. Qlaisher. " Tables of the numerical values of the Sine-integral, Cosine-
integral, and Exponential-integral. Philosophical Transactions, voL clx. pp. 367-388.
Tables of Siar, Ciar, Eijr, Ei(— a:) from x=0 to x=l at intervals of 0*01 to eighteen
places of decimals, from jr:=l to x=5 at intervals of 0*1, and from ar=5 to ^=15 at intervals of
unity to eleven places, and for a:=20 to twelve places. Ttiere are also other tables of Si or and
Ciar.
It should be mentioned also that Legendre has given in vol. iii. (p. 448) of his ^ Exercices
de Calcul Integral ' a table of
/»*/ dx da \
from ar=0 to a?=l at intervals of 0*01 to ten places of decimals ; and that a paper * by Bellavitis,
published in vol. xviii. (1874) of the Memorie delV Istituto Veneto di Scienze, Lettere ed Artiy
contains several tables connected with the logarithm-integral, the most considerable of which is
a table of liar— log log or— logo? from ^=0'5 to ar=2'8 at intervals of 001 to six places of
decimals.
Considering the function in the form li e^ or Ei x, we have for the calculation of Ei x the
formula proceeding in ascending powers of x which has been given above, viz. : —
mS m8 <m4
Ei ar=y+log a:+a:+2y3+3jg+|j^+&c.
This series, though always convergent, converges rapidly only for small values of x, and the
calculation is laborious for values of x exceeding, say, 5 or 10.
The descending series
-,. ^ (1 . 1 . 21 , 8! , 4! . . )
is semiconvergent ; and as the terms all have the same sign, it is not available for calculation as it
stands, on account of the absence of a simple rule for determining sufficiently close limits to the
error when a certain number of terms are retained.
* '' Tavole numeriche Logaritmo-integrale ossia dell* Esponenziale-intcgrale e di altri Integral! Euleriani."
n2
92 INTEODUCTION. [§ ill.
For calculating the value of £i(ar+A)— Eio: we have the formula
+
+
+
+
+
2! ^ \a:~si^)
3! ^\x x^^a»)
4! * \x a*^a» af*)
+5! ^\^;r"";r«+a^~a?« + a*/
A« /I 7 42 210 840 2520 5040 504O\
+8! ^\^i~a^+a»~ a^ + irs iB-+ a?' ~ a* j
+&C.
The law of the coefficients in the successive terms is evident : thus, ex. gr., in the term
involving V, the successive coefficients are obtained by multiplying the previous ones by 6, 5, 4,
3, 2, 1 respectively ; in the term involving A* by multiplying them by 7, 6, 5, 4, 2, 1 and so on.
As this formula is given by Taylor's theorem, the error when we stop at any term must be
equal to the value of the next term when x is replaced by x-\-dx, d being a proper fracti(m.
The sum of the first terms in the expressions involving successive powers of h is
^e»-l.-
X
and we may therefore write the formula in the form
e*+»_c» A« 1
~3\ ^ \x'~a*)
-&c (1)
The values of lia: required for Table I. (pp. 71-73) were calculated by this formula as
follows : —
Putting e'=tf and e'*^=y-\-k, so that
Ue'+*=li(y+y5:), U«'=liy,
we find that, if y and k are any two quantities, and if we put
log.y=Y,
and
log. (y+*)-log.y=log. (l +y) =Y',
§ III.] ON THE CAtCULATION OF THE FUNCTION li X. 93
then
~3Y«+3Y»
y Y" y Y^ y Y"
~8 Y«+4 Y»~4 Y*
-&C (2)
In calculating Table I. k was taken=50,000. I began the table at 500,000, and calculated
the values of the differences by this formula from 500,000 to 9,000,000. The values of Y
ranged from 13'12 to 16*01, and those of Y' (the difference between two consecutive Y's)
from 0*09 5 to 0'006. The work was carried to three decimal places ; and putting successively
y=500,000, y= 550,000, . . . y=8,950,000 it was found that
li 550,000 -li 500,000=3796-300,
li 600,000-li 550,000=3770-240,
• ••••••
li 8,950,000-li 8,900,000=3124-148,
li 9,000,000-U 8,950,000 =3123057.
The value of 11500,000 was obtained by an independent calculation and found to be
41606-289 (see pp. 94, 95) ; by adding repeatedly to this quantity the differences calculated by
the formula (2), the values of li 550,000 &c. were obtained, viz. :
li 550,000= 45402-589,
11 600,000= 49172-829,
• ' • • •
li 8,900,000=596429-070,
118,950,000=599553-218,
119,000,000=602676-275.
An independent calculation was made of li 8,900,000, and the value found to be 59642904 9,
which differs from the value obtained by successive addition of the differences by only 0'021.
The value of li 3,250,000 was also calculated independently and found to be 233687*782, the
value obtained by the successive additions being 233687-789, which differs from it by 0*007.
As the results in Table I. were required to be correct only to the nearest unit, their accuracy
was thus amply verified by the agreement of the values of li 3,250,000 and of li 8,900,000 with
those calculated independently. The values of the differences as calculated by the formula (2)
were, however, differenced as an additional verification.
It may be mentioned that the values found for 112,000,000—11 1,000,000 and li 3,000,000
—112,000,000 were respectively 70427*279 and 67935*739; these arc confirmed by Gauss's
values, which are 70427-78 and 67915-733 (Werke, vol. ii. p. 443).
The values of \ix for values of x less than 500,000 were calculated in the same manner by
means of the formula (2), though it was of course necessary to include more terms. Bessel in
the table referred to on p. 90 gave the values of lio? for a:=100,000, 200,000, 300,000, and
400,000; and the agreement of these results with those deduced by the formula (1) from the
value for ar= 500,000 to the required degree of accuracy was a sufficient verification. As an
additional verification the value of 1150,000 was deduced independently from the value of
Ei (11) given in the table (vi) referred to on p. 91 ; it was also calculated by the series proceeding
by ascending powers of ar. The value was thus found to be 5166-55.
94 INTEODTJCTION. [§ III.
I reproduce from p. 339 of vol. ii. of Bessel's ' Abhandlungen ' the table
lilOOO = 177-609658
1110,000 = 1246-137249
11100,000 = 9629-80904
11200,000 = 18036-05216
11300,000 = 26086-69223
11400,000 = 33922-62200
11500,000 = 41606-40507*
11600,000 = 49172-82711
111,000,000 = 78627-54928
and from the Philosophical Transactions, vol. clx. pp. 384 and 386, the table
Ei (1) = 1-895 117 816 36
El (2) = 4-954 234 366 00
El (3) = 9-933 832 570 62
Ei (4) = 19-630 874 470 06
Ei (5) = 40-185 275 358 80
El (6) = 85-989 762 142 44
Ei (7) = 191-504 743 335 50
Ei (8) = 440-379 899 534 84
Ei (9) = 1037-878 290 717 09
Ei(lO) = 2492-228 976 241 88
Ei(ll) = 6071-406 374 098 61
Ei(12) = 14959-532 666 397 53
Ei(13) = 37197-688 490 689 03
Ei(14) = 93192-513 633 965 37
Ei(15) = 234955-852 490 768 32
to which may be added
El (20) = 25615652-664 056 588 820
given on p. 371 of the same paper.
It will be noticed that the value of El 13 does not differ greatly from that of li 500,000 ; and
as the latter was to be used as a fundamental quantity in the calculation of the values of li x
for Table I., it seemed desirable to obtain an Independent calculation of its value to several places
of decimals by deducing it from Ei 13.
Changing the sign of h In the formula (1), and putting
«'=500,000 so that x=log. 500,000,
and
ar— A=13,
we find from (1)
U 500,000-Ei 13=^+1 5-^ ^(l-l) +&C.,
where
ar=13-122 363 377 4, A=0122 363 377 4;
* This value of li 500,000 should be 41606*28879 ; see opposite page.
§ in.] ON THE CALCULATION OP THE FUNCTION li X. 95
and including terms as far as h^ it was thus found that
Ii500,000-Ei 13=4408-600 295 8,
the last figure not necessarily being exact.
Since Ei 13=37197-6884907, we thus find
li 500,000=41606-288 786 5,
which differs from Bessel's value 41606-40507 by 0-11628.
To verify that no error had been made in the calculation of the difference li 500,000— Ei 13
another determination was made, also by means of the formula (1), taking
. ^-^*=500,000 and ar=13,
so that in this case
4r=13, A=0-122 363 377 4,
and the resulting value of the difference was found to be 4408-600 295 8, agreeing exactly with
the preceding value.
If therefore the value of Ei 13 was correct, it followed that Bessel's value of 11500,000 was
erroneous.
The value of the difference Ei 13 — li 400,000 was also calculated by means of the formula
(1), putting
^=13, ^-*=400,000,
whence
A=0-100 780 173 9.
The value was thus found to be 3275*066 565 3 ; and subtracting this from the value of £i 13
we find
11400,000=33922-621 925 4,
which differs from Bessel's value by only 0-00007. We thus obtain, therefore, a verification of
the accuracy of the values of Ei 13 and li 400,000.
Since
Ii500,000-Ei 13=4408-600 295 8,
and
Eil3~li 400,000=3275-066 565 3,
it follows by addition that
11500,000-11400,000=7683-666 861 1.
As an additional verification the value of this difference was calculated independently from
the formula (1), putting
^=500,000, ^-*=400,000,
whence
A=0-223 143 551 3.
The value was found to be 7683666 860 9, which differs from that obtained above by only 2
in the last figure.
The value of li 600,000— li 500,000 was also calculated in two ways by the formula (1), viz.
both by putting
^=600,000, ^-*= 500,000,
and by putting
^=500,000, ^+*= 600,000,
the value of h in both cases being 0182 321 556 8.
96 INTEODUOTION. [§ III.
The two calculations gave as the value of the difference
7566-538 199 8 and 7565-538 199 7
respectively.
The resulting values thus found for li 400,000 &c. are
11400,000=33922-621 925 4,
11500,000=41606-288 786 5,
11600,000=49172-826 986 2,
in which the last figure of li 600,000 is either a 2 or 3 according to which of the two values of the
difference 11600,000—11500,000 is taken. These values are, however, not necessarily correct to
the last figure ; but, from the close agreement of the results derived from the different calculations,
it is probable that the last figures are not much in error. Bessel's values, which are given to
only five places of decimals, differ from them by 0*00007, 0*11628, and 0-00012 respectively.
In connexion with Bessel's table, the following calculations, which were undertaken with
reference to Table I., may be noticed here.
The value of 11250,000-11200,000 was calculated, taking ^=200,000, and found to be
4057-9625274 ; and the value of 11300,000-11250,000 was also calculated, taking ^=300,000,
and found to be 3992-6775419. Adding these differences we find
11300,000-11200,000=8050-640 069 3.
From Bessel's values of li 200,000 and li 300,000, which extend to five places of decimals,
we find by subtraction
li 300,000-11200,000=8050-64007,
agreeing with the previous value to the last figure. Taking, therefore, Bessel's value of
li 200,000 or li 300,000, we find
11250,000=22094-01469,
which is therefore certainly correct*.
It should be mentioned that Bessel's values of 11 100,000 and li 200,000 were indirectly
verified by Gauss, who determined, as an example of his method of approximating to the values
of integrals!, the value of 11200,000-11100,000. Gauss found as his result 8406-2431211;
and by subtracting Bessel's value of 11100,000 from his value of 11200,000 we obtain the
difference 8406-24312, agreeing with Gauss's value to the last figure.
As mentioned on p. 93, it was necessary, in order to verify Table II., to make an independent
determination of the value of li j;, for some value of x near to the limit of the extent of the table.
For this purpose the value of Ei 16 was calculated by means of the series proceeding by ascending
powers oiXy and the value of li 8,900,000 was deduced from it by means of the formula (1).
By actually performing the operations of multiplication and division, it was found that
163 igS 167 1A67
16+ m +i+^" •+^7=297780-499 335 421 7,
y+log.l6H-||^V5^*...+5-^=297780-499 335 414 9,
* On p. 327 of vol. ii. of the Ahhandlungen Bessel has given the value of li 200,000 to six places of decimals, and,
using this value, we find li 250,000=22094-014680. t Werke, vol. iii. p. 196.
§ III.] ON THE CALCULATION OF THE FUNCTION li X. 97
giving, by addition and subtraction,
Eil6 = 595560-998 670 836 6,
Ei(-16)=- -000 000 006 8.
Calculating Ei (— 16) by the formula .
V / X X fl 1 . 2! 3! , „ )
Ei(-a:)=-«-'|--^4--.-^+&c.j,
we find
Ei (-16)= -0-000 000 006 6,
which diflFers from the value of Ei(— 16) found by the ascending series by only 2 in the last
place. This affords a verification of the whole calculation ; and the result
Ei 16=595560-998 670 836 6
is therefore verified, except as regards the last figure.
The value of Ei 16— li 8,900,000 was calculated by means of the formula (1), putting ar=16,
and found to be 868*0502, whence it follows that
li 8,900,000=596429-0489.
This value, as already mentioned on p. 93, differed by only 0*021 from that found by
adding up continually the differences for each group of 50,000, starting with li 500,000 ; and this
agreement afforded a sufficient verification of the accuracy of the table so formed.
The independent calculation of li 3,250,000 referred to on p. 93 was made by calculating
the difference between its value and that of Ei (15) given in the table quoted on p. 94 from the
Philosophical Transactions. Putting
^=15, ^"*= 3,250,000,
the value of A is 00058344, and the value of the difference was found to be 1268*071, giving
li 3,250,000=233687-782,
which differs from the value found in the calculation of Table I. by only 0-007.
The values of li 10,000,000 and li 100,000,000 were calculated in the following manner by
means of formulae given by Bessel {Abhandltrngeriy voL ii. p. 338).
Bessel has there shown that
• |wi ' m— 1 ^ m— 2 ^m— r+1 j
"T"^ |i»«"*"(m-l)«"r(m-2)« ' • •"T"(w-r-|-l)«j
+^' '|^+(m-l)s+(»t-2)« • • '+(m-r+l)«/
4-
I • • • • • a ,
and for the case a =10 he has given a table of the values of the coefficients B', B" This
table contains the value of nB^"^ to eleven places of decimals and of log^^ B^*^ to ten places as far
as 71=29.
98 INTEODFCTION. [^ HI-
In the case r=l, the fonnula becomes
li a«=li a->+B' ^ +B" ^+BI"^+&c.,
from which, putting m=7 and w=8, and working to two places of decimals, we obtain
U10^-lilO«=: 586,290-85,
li 10«-li 107=5,097,290-98.
Since li 10^=U 1,000,000=78627-55, we thus find
li 10,000,000= 664,918-40,
li 100,000,000=5,762,209-38,
in which results of course the last figure is not necessarily correct.
As a verification the value of 1110,000,000 was deduced from Eil6 by means of the
formula (1). Taking ar=16, the difference was found to be 69357-41, giving 1110,000,000
=664918*41, which differs from the value found above by only a unit in the last place. This
agreement affords also a verification of the accuracy of Bessers coefficients.
It is interesting to give the values of the successive terms in the descending series for
li 100,000,000. The series is
F+^«+ oH* + a^ +-ir+«c.;
and denoting by [n] the nth term, ^^ — ^r~"> ^^^^ the values of the terms are as follows : —
c; Yf tu
f^i^, ^ » '
^ucii tiic vaiuca ui
tiic; tc
«•=
=100,000,000
[1]
5428681-02,
[13]
1-70,
[2]
294705-78,
[14]
1-20,
[3]
31997-27,
[15]
0-91,
[4]
5211-09,
[16]
0-74,
[6]
1131-57,
[17]
0-65,
[6]
307-15,
[18]
0-60,
[7]
10004,
[19]
0-58,
[8]
38-02,
[20"
0-60,
[9]
16-52,
[21]
0-65,
[10]
8-07,
[22]
0-74,
[11]
4-38,
[23]
0-80,
[12]
2-62,
i
&c.
is 1^'**. which =0-51
i ! and tha Aum
nf th
=5,762,209'33, which differs from the correct value by only 0-05.
When ^'=10,000,000, the least term is -^, which=0-63 ; and the sum of the preceding
terms=664917-92, which differs from the correct value by 0*5.
^ III.] ON THE CALCULATION OF ETEM ANN'S FORMULA. 99
On the Calculation of Siemann^s Formula.
Riemann's formula is
1 I
in which the general term is - li jr", where n denotes any number not divisible by a squared factor,
that is to say, any number of the form abc . . . where a^by c, . . , are different primes. The sign
of the term is positive when the number of the prime factors a, by c is even, and negative when
the number is uneven.
The first term li x was calculated for Table I. (pp. 71-73) in the manner already explained.
The other terms were calculated by means of Stenberg's table in the following manner.
The portion of Stenberg's table in which the arguments are greater than unity contains the
values of li 10* from a=0 to a=3-5. Between a:=l and ar=3-6, differences as far as the seventh
order are given.
The terms to be calculated may be written in the form
-ililO*^^»'^-ililO*^^«^'^--iU10*^^«'^+ililO*^^^^-|lilO*^^»^+Ti5Ul^^
The first step was to form the values of
ilogioo:, ilogio^, i logic ar, ilogio^r, |logio^, i^ logic a:, Alogio^,
from a:= 50,000 to ar= 9,000,000 at intervals of 50,000. These were then taken as arguments,
and the corresponding values of li 10* were obtained by interpolation from Stenberg's table.
As examples of the work I take the extreme cases of dr= 50,000 and ar= 9,000,000.
In the case of a;= 50,000,
logj;=4-69897, i log 4r= 0-78316,
ilog4r=:2*34949, |logar=0-67128,
iloga:=l-56632, ^^ log ar= 0-46990,
iloga:=0-93979, -^r log ar=: 0-4271 8;
and, by interpolation from Stenberg's tables,
li 1 ()»•"•*•= 54-601, U10'''""=4-261,
lil0i««»=14-974, lilO»'«"»= 3-439,
lilO»*»»= 5-686, lil0»-**«'=2-118,
U10<»-*""=l-850;
whence the values of the snccessive terms in Biemann's formula are
-27-300, -4-991, -1-117, +0-710, -0-491, +0-212, -0168.
The sum of the negative terms =34*067 ; the sum of the positive terms =0-922 ; whence
the final result =—33-145.
In the case of a? =9,000,000,
logx=6-95424, i log r= 1-15904,
i log ar= 3-47712, | log a:= 0-99346,
ilogar=2-31808, t^^ log a?= 0-69542,
ilog:F=l-39085, tV log a:= 0-63220,
o2
100 INTEODUCnON. [§ III-
and
U 10'**"^= 442-758, li 10^-^«^= 7-940,
li 10231808^ 51-698, lilO^~^«= 6-100,
lilO^-^^= 11-387, lil0'«^=3-609,
U10^«^=:3-170;
whence the values of the terms are
-221-379, -17-233, -2-277, +1323, -0871, +0-361, -0-288.
The sum of the negative terms =242-048 ; the sum of the positive terms =1684 ; whence
the final result =-240-364.
The values of the terms
«.^li^4«^lia:*-|liar*+ilia?* . . .-■ Arli^"^
were calculated throughout in this manner, and the results verified by differences. It will be
noticed that the value of the last term included, viz. -j^liar^, is 0*168 for ar=50,000 and 0-288
for 0^=9,000,000. By stopping therefore at this term in the calculation of the formula we obtain
results which, though not necessarily true to the nearest unit, are probably never so much as a
unit in error except by contraction.
In order to examine the effects of the subsequent portion of the series, the values of the next
ten terms, viz.
-Alia:^+Aliar^* + i\ lia;^--iVli^'''-TVli^'''
were calculated for one value in each million, viz. for 4:= 1,000,000, ar=2,000,000, . . .
x= 9,000,000, with the following results : —
*.
Negative terms.
PosiiiTe terms
1,000,000
0-347
0-350
2,000,000
0-377
0-380
3,000,000
0-394
0-397
4,000,000
0-407
0-410
5,000,000
0-416
0-419
6,000,000
0-424
0-427
7,000,000
0-431
0-434
8,000,000
0-437
0-440
9,000,000
0-441
0-445
The total correction due to these ten terms therefore never exceeds 0*004 ; and it is evident
that they cannot appreciably affect the accuracy of results which were only required to be
correct to the nearest unit.
Continuing the calculation till the terms become insensible to the third decimal place, which
occurs at the term -^ li x^ for 4r= 1,000,000, and at the term -^ li or^ for ar= 9,000,000, we obtain
the following values for the sums of the terms between -^ li x^ and the insensible terms : —
§ III.] ON THE CALCULATION OF EIEMANN'S FORMULA. 101
X,
Negative terms.
Positive terms
1 ,000,000
0020
0012
2,000,000
0026
0019
3,000,000
0031
0025
4,000,000
0033
0029
5,000,000
0036
0032
6,000,000
0038
0034
7,000,000
0040
0037
8,000,000
0042
0-038
9,000,000
0044
0040
The greatest value of this additional correction is therefore —0008, and the total value
of all the terms from the term -j^liar^*^ to the term that becomes insensible never exceeds
-0005.
As an example of the gradual diminution of the terms I give the values of the successive
1 1
terms in the case of ar= 9,000,000. The notation \n\ is used to denote the term ^li^-
d?= 9,000,000.
Negative terms.
Pontire tenns.
[2] =
221-379
[6]
=
1-323
3] =
17-233
[10]
0-361
[6] =
2-277
[14]
=z=
0-163
7] =
0-871
[15]
=
0-139
11] =
0-288
[21]
=
0069
[13] =
0-195
[22]
=
0-052
[17] =
0-102
[26]
=
0-031
;i9] =
0-077
[33]
=
0-012
;23] r=
0-045
[34;
=:
0010
'29] =
0-021
:35:
=
0-009
[30] =
0-019
[38"
=
0-005
[31] =
0-016
[39]
=:;
0-004
[37] =
0-006
;41] =
0-002
;42] =
0-001
■43] =
0-000
It is to be noticed, however, that the terms become insensible, not in consequence of the
diminution of the coefficients, but because the point where the function li a: vanishes is reached.
After this point is passed, the terms begin to increase. The function li^ vanishes when
ar=l*451369 . . . ; and therefore lilO* vanishes when a=0'161777 ... As we approach this
value as argument the values of the successive terms in Riemann's formula diminish, but if
we continue the calculation further they increase at first and then diminish to zero very slowly.
102 INTRODUCTION. [§111.
The following table shows the values of the terms from -^ li x^ to g-^ ^^ ^'^ ^^^ a:= 9,000,000,
and forms a continaation of the series of values just given. It will be observed that, as the
function li a; changes sign when a:=l'451369. .. and is negative for arguments less than this
quantity, the values of the ' negative ' terms (i. e. the terms occurring with negative coefficients
in Riemann's formula) are positive, and those of the * positive ' terms negative.
x=
=9,000,000.
Negative
terms.
Positive term».
[47] =
-0003
;46)
—
-0002
"53] =
-0005
;5i;
=
-0-005
"59] =
-0007
;55]
—
-0006
[61] =
-0008
pf'.
—
-0-007
;66] =
-0009
[58]
=
-0-007
[67] =
-0 009
[62;
=
-0008
70] =
-0-009
65;
=
-0009
;7i] =
-0009
;69;
=
-0009
[73] =
-0010
74'
=r
-0010
78] =
-0010
[77;
=
-0010
79] =
-0010
;82]
—
-0010
;83] =
-0-010
[85"
=
-0010
"89] =
-0011
[86]
=
-0011
"97] =
-0011
[87;
=
-0011
;9i;
=
-0011
;93;
—
-0011
;94]
:=
-0011
;95]
=
-0-011
Up to this point the terms are still increasing, though so slowly that for some distance their
values are nearly constant to three places of decimals. The following table, which contains the
numerical values of selected terms separated by intervals of about 100, serves to show the
extreme slowness with which the terms diminish after passing their maximum : —
[101]
0-011
;60i;
0005
;20i;
0-009
[70i;
004
;30i]
0-008
;802;
0004
;40i;
0007
;90i;
0004
■501;
0006
;iooi]
0003
As the value of Riemann's series was only required to the nearest unit, it appeared therefore
that it was unnecessary to take account of the terms beyond -j^ li a;^\ and the numbers given in
Table V. (pp. 84-86) are those derived from the calculation of the series as far as this term only.
The preceding results also show how laborious would be the calculation of the series term by
term if greater accuracy were attempted.
The manner in which li 10,000,000 and li 100,000,000 were calculated has been already
explained on pp. 97 and 98. The other terms occurring in Riemann's formula were calculated
for ii-= 10,000,000 and ;r =100,000,000 by interpolation from Stenberg's table, except in the
case of \ia^ for a:= 100,000,000, which =li 1 0,000 =li 10*, and is therefore beyond the range of
§ III.] ON THE CALCULATION OP MBMANN'S FOEMTJLA. 103
the table. The value of li 10,000 is, however, given by Bessel {Ahhandlungen^ voL ii. p. 339), and
his value viras used.
I give below the values of the terms in the case of j;= 100,000,000 as far as the point where
the minimum term is reached ; of course the subsequent terms increase and then slowly diminish,
as in the case of ar= 9,000,000.
a:= 100,000,000.
Negati
ve terms.
Poeitive tenmi.
[2]
623069
[6]
1-735
[3]
31-997
[10]
0-439
[5]
3158
[14]
0-198
[7]
1106
[15]
0168
[11]
0-349
[21]
0-074
[13]
0-235
[22]
0-066
[17]
0-125
[26]
0-042
[19]
0-095
[33]
0-019
[23"
0-059
[34]
0-017
[29]
0031
[35]
0-015
[30]
0-027
[38]
0011
[31]
0024
[39]
0-009
[37]
0012
[46]
0-002
[41]
0-007
[42]
0-006
[43]
0005
[47]
0-001
It is easy to see that in all cases the terms ultimately decrease to zero. This may be
1
proved either by finding the limiting value of — for w=oo , which, differentiating the numerator
and denominator,
I
=the limit of — ^ =0 ;
n
or by considering the series which defines lijr; for the term which becomes infinite as n increases
I
is log log a:~, and
- log log4ri=-(log -+log log ^),
which is zero when n is infinite.
FACTOR TABLE
FOR THE
SIXTH MILLION
' »
6000000
[500]
500
101
04107
10
13 16
19
22 25
28 1 31
34
37 140
43 146
49 152
55
58
61
64
67 170
73(76
79 t 82 1 85 1
w
• •
—
7
—
—
II
109
*3
7
—
13
^
311
7
II
—
43
15'
J^
—
7
—
'3
»53>
II
163
07
• ••
7
43
1039
II
—
»3
—
7
—
37
4»
—
19
7
47
29
1291
»3
139
787
7
71
—
»3
II
—
97
7
11
• • •
»3
821
73
7
97
—
—
37
II
7
—
19
13
41
1123
7| *4i
II
—
—
—
»5'
7
17
13
43
13
• • •
—
II
—
31
19
1009
7
1553
»79
103
709
II
7
53
17
691
^
»1
41
7
73
2087
II
13
37
7
—
17
•
127
331
433
7
47
>9
2099
II
13
7
1097
—
»3
29
1 109
7
II
137
13
31
■^
7
37
53
17
—
19
• • B
a39
163
61
—
7
13
—
2y
II
127
7
83
19
17
—
79
7
II
—
223
—
7
—
4'
23
• ■ft
349
7
3«
II
—
23
7
17
283
857
■—
13
7
71
881
7
•"^
»9
II
»3
29
• • •
227
7
a3
II
61
—
107
—
7
»3
383
—
17
67
II
7
3«
139
47
—
—
7
1583
277
II
-^
19
—
31
• • •
•3
17
29
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